Find the Volume of Each Prism. 1. SOLUTION

Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 12-4 Volumes of Prisms and Cylinders 3 396 in

Find the volume of each prism. 3. the oblique rectangular prism shown at the right

1. SOLUTION: SOLUTION: The volume V of a prism is V = Bh, where B is the If two solids have the same height h and the same area of a base and h is the height of the prism. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

3 The volume is 108 cm .

ANSWER: ANSWER: 108 cm3 26.95 m3

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have 2. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional SOLUTION: area are same. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right 5. SOLUTION: eSolutions Manual - Powered by Cognero Page 1 SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 3 206.4 ft

6. SOLUTION: ANSWER: If two solids have the same height h and the same 3 26.95 m cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder 4. an oblique pentagonal prism with a base area of 42 and an oblique one of the same height and cross

square centimeters and a height of 5.2 centimeters sectional area are same. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a

height of 5.1 centimeters ANSWER: SOLUTION: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 3 5. 1025.4 cm SOLUTION: 8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION:

ANSWER: 3 206.4 ft ANSWER: 3 410.1 in

9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 6. 7.5 gallons, how many gallons will it take to fill the SOLUTION: lap pool? If two solids have the same height h and the same A 4000 cross-sectional area B at every level, then they have B 6400 the same volume. So, the volume of a right cylinder C 30,000 and an oblique one of the same height and cross D 48,000 sectional area are same. SOLUTION:

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) ANSWER: = 48,000.

3 1357.2 m Therefore, the correct choice is D.

7. a cylinder with a diameter of 16 centimeters and a ANSWER: height of 5.1 centimeters D SOLUTION: CCSS SENSE-MAKING Find the volume of each prism.

10. SOLUTION: ANSWER: 3 The base is a rectangle of length 3 in. and width 2 in. 1025.4 cm The height of the prism is 5 in.

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches

SOLUTION:

ANSWER: 30 in3

ANSWER: 3 410.1 in

9. MULTIPLE CHOICE A rectangular lap pool 11. measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds SOLUTION: 7.5 gallons, how many gallons will it take to fill the The base is a triangle with a base length of 11 m and lap pool? the corresponding height of 7 m. The height of the A 4000 prism is 14 m. B 6400 C 30,000 D 48,000 SOLUTION:

ANSWER: 3 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. 12. ANSWER: SOLUTION: D The base is a right triangle with a leg length of 9 cm CCSS SENSE-MAKING Find the volume of and the hypotenuse of length 15 cm. each prism. Use the Pythagorean Theorem to find the height of the base.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

The height of the prism is 6 cm.

ANSWER: 30 in3

ANSWER: 324 cm3

11. SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the 13. prism is 14 m. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 3 The volume V of a prism is V = Bh, where B is the 539 m area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

12. ANSWER: 3 SOLUTION: 58.14 ft The base is a right triangle with a leg length of 9 cm 14. an oblique hexagonal prism with a height of 15 and the hypotenuse of length 15 cm. centimeters and with a base area of 136 square centimeters Use the Pythagorean Theorem to find the height of the base. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The height of the prism is 6 cm. ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches ANSWER: SOLUTION: 3 324 cm If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

13. SOLUTION: If two solids have the same height h and the same

cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 3 area are same. 1534.25 in CCSS SENSE-MAKING Find the volume of The volume V of a prism is V = Bh, where B is the each cylinder. Round to the nearest tenth. area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

16. ANSWER: SOLUTION: 58.14 ft3 r = 5 yd and h = 18 yd

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 3 the same volume. So, the volume of a right prism and 1413.7 yd an oblique one of the same height and cross sectional area are same.

17. SOLUTION:

r = 6 cm and h = 3.6 cm. ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches

SOLUTION: If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 3 the same volume. So, the volume of a right prism and 407.2 cm an oblique one of the same height and cross sectional area are same.

18. SOLUTION:

r = 5.5 in. ANSWER: 1534.25 in3 Use the Pythagorean Theorem to find the height of the cylinder. CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. SOLUTION: r = 5 yd and h = 18 yd

Now you can find the volume.

ANSWER: 3 1413.7 yd

ANSWER: 3 823.0 in 17. SOLUTION: r = 6 cm and h = 3.6 cm.

19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 3 area are same. 407.2 cm r = 7.5 mm and h = 15.2 mm.

18.

SOLUTION: r = 5.5 in. ANSWER: Use the Pythagorean Theorem to find the height of 2686.1 mm3 the cylinder. 20. PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER: ANSWER: 3 3 823.0 in 2740.5 in

21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 19. centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space SOLUTION: in the box? If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm. SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

ANSWER: 3 2686.1 mm

20. PLANTER A planter is in the shape of a ANSWER: rectangular prism 18 inches long, inches deep, 521.5 cm3 and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, below the top? and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that SOLUTION: are 2 feet tall to hold enough sand for one contestant.

The planter is to be filled inches below the top, so What should the diameter of the cylinders be? SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3 2740.5 in Therefore, the diameter of the cylinders should be 21. SHIPPING A box 18 centimeters by 9 centimeters about 2.52 ft. by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 ANSWER: centimeters and a height of 15 centimeters, as shown 2.52 ft at the right. What is the volume of the empty space in the box? Find the volume of the solid formed by each net.

SOLUTION: 23. The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

ANSWER: 521.5 cm3 The height of the prism is 20 cm. 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, The volume V of a prism is V = Bh, where B is the and 10 cubic feet of sand. To transport the correct area of the base, h is the height of the prism. amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r. ANSWER:

3934.9 cm3

Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER: 24. 2.52 ft SOLUTION: The circular bases at the top and bottom of the net Find the volume of the solid formed by each indicate that this is a cylinder. If the middle piece net. were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

23. If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and The middle piece of the net is the front of the solid. an oblique one of the same height and cross sectional The top and bottom pieces are the bases and the area are same. pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm. ANSWER: 3 48.9 m The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism. 25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

ANSWER: 3934.9 cm3

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

24. SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique. The height of the new can will be 35.1 cm. The radius is 1.8 m, the height is 4.8 m, and the slant ANSWER: height is 6 m. 35.1 cm If two solids have the same height h and the same 26. CHANGING DIMENSIONS A cylinder has a cross-sectional area B at every level, then they have radius of 5 centimeters and a height of 8 centimeters. the same volume. So, the volume of a right prism and Describe how each change affects the volume of the an oblique one of the same height and cross sectional cylinder. area are same. a. The height is tripled.

b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

ANSWER: 3 48.9 m a. 25. FOOD A cylindrical can of baked potato chips has a When the height is tripled, h = 3h. height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

When the height is tripled, the volume is multiplied by 3. SOLUTION: b. When the radius is tripled, r = 3r. The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. When the height and the radius are tripled, the a. The height is tripled. volume is multiplied by 27. b. The radius is tripled. c. Both the radius and the height are tripled. d. When the dimensions are exchanged, r = 8 and h d. The dimensions are exchanged. = 5 cm. SOLUTION:

Compare to the original volume. a. When the height is tripled, h = 3h.

The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9.

c. 3 When the height is tripled, the volume is multiplied by The volume is multiplied by 3 or 27.

3. d. The volume is multiplied by b. When the radius is tripled, r = 3r.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 So, when the radius is tripled, the volume is multiplied pounds, what is the soil’s bulk density? b. Assuming by 9. that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound c. When the height and the radius are tripled, r = 3r per square inch is desirable for root growth? Explain. and h = 3h. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION:

When the height and the radius are tripled, the volume is multiplied by 27. a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. d. When the dimensions are exchanged, r = 8 and h = 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. Compare to the original volume. W = 20 – 5 = 15 lbs.

The soil density is thus:

The volume is multiplied by .

ANSWER: 3 3 a. The volume is multiplied by 3. b. 0.0018 lb/in is close to 0.0019 lb/in so the plant 2 should grow fairly well. b. The volume is multiplied by 3 or 9.

3 c. The volume is multiplied by 3 or 27. c.

d. The volume is multiplied by

3 a. If the weight of the container with the soil is 20 a. 0.0019 lb / in pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming b. The plant should grow well in this soil since the that all other factors are favorable, how well should a bulk density of 0.0019 lb / in3 is close to the desired plant grow in this soil if a bulk density of 0.018 pound 3 per square inch is desirable for root growth? Explain. bulk density of 0.0018 lb / in .

c. If a bag of this soil holds 2.5 cubic feet, what is its c. 8.3 lb weight in pounds? SOLUTION: Find the volume of each composite solid. Round to the nearest tenth if necessary.

a. First calculate the volume of soil in the pot. Then 28. divide the weight of the soil by the volume. SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus: ANSWER: 225 cm3

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular ANSWER: prisms. 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary. ANSWER: 3 120 m

28.

SOLUTION: 30. The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm SOLUTION: and the height is 11 cm. The base of the other prism The solid is a combination of a rectangular prism and is 4 cm by 3 cm and the height is 5 cm. two half cylinders.

ANSWER: 225 cm3 ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the 29. volume of the rubberized material that makes up the holder? SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a

radius of and a height of 11.5 cm. ANSWER: The empty space used to keep the can has a radius 3 120 m of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

30. Therefore, the volume of the rubberized material is SOLUTION: 3 The solid is a combination of a rectangular prism and about 304.1 cm .

two half cylinders. ANSWER:

3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height?

SOLUTION:

ANSWER: 260.5 in3

the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the

holder? ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is The radius is 6. Find the volume. 3 about 304.1 cm . ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION: ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Find the volume.

Use the surface area formula to solve for r.

ANSWER: 3 576 in

SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume?

SOLUTION: Use the surface area formula to find the length of the ANSWER: base of the prism. 3,190,680.0 cm3

Find the volume.

SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ANSWER: ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of 3 576 in the solid is the sum of the volumes of the two prisms. 35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter.

SOLUTION: The volume will be the highest when the diameter is ANSWER: 3 95 cm and will be the lowest when it is 30 cm.That is 1575 ft when the radii are 47.5 cm and 15 cm respectively. 37. CHANGING DIMENSIONS A soy milk company Find the difference between the volumes. is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the same. What will be the height of the new containers?

ANSWER: SOLUTION: 3 3,190,680.0 cm Find the volume of the original container.

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool. The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height ANSWER: of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms. 38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following ANSWER: equation for volume: 3 1575 ft

The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height.

SOLUTION: Find the volume of the original container.

If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

ANSWER: For any rectangular container, the volume equation is:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds Choose numbers for any two of the dimensions and 1 cup. Be sure to include the dimensions in each we can solve for the third. Let l = 2.25 in. and w = drawing. (1 cup ≈ 14.4375 in3) 2.5 in. SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

SOLUTION: The base of the prism can be divided into 5 congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: Sample answers: ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

The total cost for the second contractor is about .

SOLUTION: Therefore, the second contractor is a less expensive The base of the prism can be divided into 5 option. congruent triangles of a base 8 cm and the ANSWER: corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height Because 2.96 yd3 of concrete are needed, the 10 cm. Find the base area of each triangular prism. second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this Therefore, the volume of the pentagonal prism is problem, you will investigate right and oblique cylinders . a. GEOMETRIC ANSWER: Draw a right cylinder and an oblique cylinder with a height of 10 meters and a 3 1100 cm ; Each triangular prism has a base area of diameter of 6 meters. 2 or 22 cm and a height of 10 cm. b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume 40. PATIOS Mr. Thomas is planning to remove an old greater than, less than, or equal to the volume of patio and install a new rectangular concrete patio 20 the cylinder? Explain. feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second c. ANALYTICAL Describe which change affects contractor bid $500 per cubic yard for the new patio the volume of the cylinder more: multiplying the and $700 for removal of the old patio. Which is the height by x or multiplying the radius by x. Explain. less expensive option? Explain. SOLUTION: SOLUTION: a. The oblique cylinder should look like the right Convert all of the dimensions to yards. cylinder (same height and size), except that it is pushed a little to the side, like a slinky. 20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

b. Find the volume of each.

The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option. The volume of the square prism is greater. c. Do each scenario. ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

b. VERBAL A square prism has a height of 10 Assuming x > 1, multiplying the radius by x makes meters and a base edge of 6 meters. Is its volume 2 greater than, less than, or equal to the volume of the volume x times greater. the cylinder? Explain. For example, if x = 0.5, then x2 = 0.25, which is less c. ANALYTICAL Describe which change affects than x. the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. ANSWER: a. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 2 b. Find the volume of each. m has an area of 9π or 28.3 m . Since the heights are the same, the volume of the square prism is greater.

42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5

The volume of the square prism is greater. units. Is either of them correct? Explain your reasoning. c. Do each scenario.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the Assuming x > 1, multiplying the radius by x makes base. 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less than x.

ANSWER: ANSWER: a. Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

b. Greater than; a square with a side length of 6 m a. rectangular prism 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights b. square prism are the same, the volume of the square prism is greater. c. triangular prism with a right triangle as the base

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1. SOLUTION: 42. CCSS CRITIQUE Franciso and Valerie each 3 calculated the volume of an equilateral triangular The volume of the can is 20π in . It takes three full prism with an apothem of 4 units and height of 5 cans to fill the container, so the volume of the units. Is either of them correct? Explain your 3 container is 60π in . reasoning.

a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct. 3 by 5 by 4π Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- b. Choose some basic values for 2 of the sides, and half the apothem multiplied by the perimeter of the then determine the third side. Base: 5 by 5. base.

ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct. 5 by 5 by 43. CHALLENGE A cylindrical can is used to fill a

container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the c. Choose some basic values for 2 of the sides, and container if it is each of the following shapes. then determine the third side. Base: Legs: 3 by 4.

a. rectangular prism

b. square prism

c. triangular prism with a right triangle as the base

3 by 4 by 10π

ANSWER: Sample answers:

SOLUTION: a. 3 by 5 by 4π

in3 The volume of the can is 20π . It takes three full b. 5 by 5 by cans to fill the container, so the volume of the 3 container is 60π in . c. base with legs measuring 3 in. and 4 in., height 10π in. a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? 3 by 5 by 4π SOLUTION:

Sample answer: The nursery means a cubic yard, b. Choose some basic values for 2 of the sides, and 3 then determine the third side. Base: 5 by 5. which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a 5 by 5 by volume of 50 cubic centimeters. SOLUTION: c. Choose some basic values for 2 of the sides, and Choose 3 values that multiply to make 50. The then determine the third side. Base: Legs: 3 by 4. factors of 50 are 2, 5, 5, so these are the simplest values to choose.

Sample answer:

3 by 4 by 10π ANSWER: ANSWER: Sample answer: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 46. REASONING Determine whether the following 10π in. statement is true or false. Explain. Two cylinders with the same height and the same 44. WRITING IN MATH Write a helpful response to lateral area must have the same volume. the following question posted on an Internet SOLUTION: gardening forum. True; if two cylinders have the same height (h1 = h2) I am new to gardening. The nursery will deliver a and the same lateral area (L1 = L2), the circular truckload of soil, which they say is 4 yards. I bases must have the same area. know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: The radii must also be equal. Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your ANSWER: garden in cubic feet and divide by 27 to determine True; if two cylinders have the same height and the the number of cubic yards of soil needed. same lateral area, the circular bases must have the 2 same area. Therefore, πr h is the same for each 45. OPEN ENDED Draw and label a prism that has a cylinder. volume of 50 cubic centimeters. SOLUTION: 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder Choose 3 values that multiply to make 50. The similar? How are they different? factors of 50 are 2, 5, 5, so these are the simplest values to choose. SOLUTION: Both formulas involve multiplying the area of the Sample answer: base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the ANSWER: area of the base by the height. The base of a prism is Sample answer: a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr .

48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs

measuring 8 centimeters and 15 centimeters. What is 46. REASONING Determine whether the following the height of the prism? statement is true or false. Explain. A 34.5 cm Two cylinders with the same height and the same B 23 cm lateral area must have the same volume. C 17 cm D SOLUTION: 11.5 cm SOLUTION: True; if two cylinders have the same height (h1 = h2)

and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: The radii must also be equal. B ANSWER: 49. A cylindrical tank used for oil storage has a height True; if two cylinders have the same height and the that is half the length of its radius. If the volume of same lateral area, the circular bases must have the 3 2 the tank is 1,122,360 ft , what is the tank’s radius? same area. Therefore, πr h is the same for each F cylinder. 89.4 ft G 178.8 ft 47. WRITING IN MATH How are the formulas for H 280.9 ft the volume of a prism and the volume of a cylinder J 561.8 ft similar? How are they different? SOLUTION: SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

48. The volume of a triangular prism is 1380 cubic ANSWER: centimeters. Its base is a right triangle with legs F measuring 8 centimeters and 15 centimeters. What is the height of the prism? 50. SHORT RESPONSE What is the ratio of the area A 34.5 cm of the circle to the area of the square? B 23 cm C 17 cm D 11.5 cm SOLUTION: SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

ANSWER: 51. SAT/ACT A county proposes to enact a new 0.5% B property tax. What would be the additional tax amount for a landowner whose property has a 49. A cylindrical tank used for oil storage has a height taxable value of $85,000? that is half the length of its radius. If the volume of A $4.25 3 the tank is 1,122,360 ft , what is the tank’s radius? B $170 F 89.4 ft C $425 G 178.8 ft D $4250 H 280.9 ft E $42,500 J 561.8 ft SOLUTION: SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. ANSWER: F SOLUTION:

50. SHORT RESPONSE What is the ratio of the area The lateral area L of a regular pyramid is ,

of the circle to the area of the square? where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the SOLUTION: Pythagorean Theorem to find the slant height. The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax

amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 D $4250 E $42,500

SOLUTION: Find the perimeter and area of the equilateral Find the 0.5% of $85,000. triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. The perimeter is P = 3 × 10 or 30 feet.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 212.1 ft2; 255.4 ft2

53. SOLUTION:

The lateral area L of a regular pyramid is ,

The perimeter is P = 3 × 10 or 30 feet. where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the area of the base B is ft2. 2 So, the lateral area of the pyramid is 126 cm . Find the lateral area L and surface area S of the regular pyramid. The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is about 212.1 ft2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 54. 212.1 ft2; 255.4 ft2 SOLUTION: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

53. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant

height is 9 cm. Use a trigonometric ratio to find the measure of the apothem a.

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular Find the lateral area and surface area of the pyramid is , where pyramid. L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 758.9 in2. 54.

SOLUTION: ANSWER: The pyramid has a slant height of 15 inches and the 472.5 in2; 758.9 in2 base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the 55. BAKING Many baking pans are given a special angle formed in the triangle below is 30°. nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

SOLUTION: The area that needs to be coated is the sum of the

lateral area and one base area. Use a trigonometric ratio to find the measure of the Therefore, the area that needs to be coated is 2(13 + 2 apothem a. 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the Find the lateral area and surface area of the diameter. pyramid. SOLUTION:

So, the lateral area of the pyramid is 472.5 in2.

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 Therefore, the surface area of the pyramid is about square centimeters. 2 758.9 in . SOLUTION:

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm SOLUTION: The area that needs to be coated is the sum of the 58. The area of a circle is 191 square feet. Find the lateral area and one base area. radius. Therefore, the area that needs to be coated is 2(13 + 2 SOLUTION: 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION: ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches. SOLUTION:

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters.

SOLUTION: ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

The diameter of the circle is about 11.4 m. 60. SOLUTION: ANSWER: The area A of a kite is one half the product of the 11.4 cm lengths of its diagonals, d1 and d2. 58. The area of a circle is 191 square feet. Find the radius. d1 = 12 in. and d2 = 7 + 13 = 20 in. SOLUTION:

ANSWER: 2 120 in

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches. 61. SOLUTION: SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: ANSWER: 2 9.3 in. 378 m Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

Find the volume of each prism.

ANSWER: 3 396 in 1. 3. the oblique rectangular prism shown at the right SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 3 an oblique one of the same height and cross sectional The volume is 108 cm . area are same.

ANSWER: 108 cm3

ANSWER: 26.95 m3

2. 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: The volume V of a prism is V = Bh, where B is the SOLUTION: area of a base and h is the height of the prism. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: ANSWER: 3 3 396 in 218.4 cm

3. the oblique rectangular prism shown at the right Find the volume of each cylinder. Round to the nearest tenth.

SOLUTION: 5. If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have

the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 206.4 ft3

ANSWER: 26.95 m3

4. an oblique pentagonal prism with a base area of 42 6. square centimeters and a height of 5.2 centimeters SOLUTION: SOLUTION: If two solids have the same height h and the same If two solids have the same height h and the same cross-sectional area B at every level, then they have cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder the same volume. So, the volume of a right prism and and an oblique one of the same height and cross an oblique one of the same height and cross sectional sectional area are same. area are same.

ANSWER: ANSWER: 12-4 Volumes of Prisms and Cylinders 3 3 218.4 cm 1357.2 m

Find the volume of each cylinder. Round to the 7. a cylinder with a diameter of 16 centimeters and a nearest tenth. height of 5.1 centimeters SOLUTION:

5. SOLUTION:

ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches

ANSWER: SOLUTION: 206.4 ft3

6. ANSWER: 3 SOLUTION: 410.1 in If two solids have the same height h and the same 9. MULTIPLE CHOICE A rectangular lap pool cross-sectional area B at every level, then they have measures 80 feet long by 20 feet wide. If it needs to the same volume. So, the volume of a right cylinder be filled to four feet deep and each cubic foot holds and an oblique one of the same height and cross 7.5 gallons, how many gallons will it take to fill the

sectional area are same. lap pool?

A 4000 B 6400 C 30,000 D 48,000

SOLUTION: ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a Each cubic foot holds 7.5 gallons of water. So, the height of 5.1 centimeters amount of water required to fill the pool is 6400(7.5) = 48,000. SOLUTION: Therefore, the correct choice is D.

ANSWER: D CCSS SENSE-MAKING Find the volume of each prism. eSolutionsANSWER: Manual - Powered by Cognero Page 2 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 10. 7.4 inches SOLUTION: SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

ANSWER: 3 410.1 in ANSWER: 30 in3 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 11. C 30,000 SOLUTION: D 48,000 The base is a triangle with a base length of 11 m and SOLUTION: the corresponding height of 7 m. The height of the prism is 14 m.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000. ANSWER: 3 Therefore, the correct choice is D. 539 m ANSWER: D CCSS SENSE-MAKING Find the volume of each prism. 12. SOLUTION: 10. The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. Use the Pythagorean Theorem to find the height of The height of the prism is 5 in. the base.

ANSWER: 3 30 in

The height of the prism is 6 cm.

11. SOLUTION: ANSWER: The base is a triangle with a base length of 11 m and 324 cm3 the corresponding height of 7 m. The height of the prism is 14 m.

13. SOLUTION: ANSWER: 3 If two solids have the same height h and the same 539 m cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the 12. area of a base and h is the height of the prism.

SOLUTION: 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is The base is a right triangle with a leg length of 9 cm

and the hypotenuse of length 15 cm.

ANSWER: Use the Pythagorean Theorem to find the height of the base. 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional

The height of the prism is 6 cm. area are same.

ANSWER: ANSWER: 3 324 cm 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same 13. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and SOLUTION: an oblique one of the same height and cross sectional If two solids have the same height h and the same area are same. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. ANSWER: 2 3 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is 1534.25 in

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square 16.

centimeters SOLUTION: SOLUTION: r = 5 yd and h = 18 yd If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional

area are same.

ANSWER: 3 1413.7 yd

ANSWER: 3 2040 cm 17. 15. a square prism with a base edge of 9.5 inches and a SOLUTION:

height of 17 inches r = 6 cm and h = 3.6 cm. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 407.2 cm3

ANSWER: 1534.25 in3 18. CCSS SENSE-MAKING Find the volume of SOLUTION: each cylinder. Round to the nearest tenth. r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder. 16. SOLUTION: r = 5 yd and h = 18 yd

ANSWER:

3 1413.7 yd Now you can find the volume.

17.

SOLUTION: r = 6 cm and h = 3.6 cm. ANSWER: 3 823.0 in

ANSWER: 19. 407.2 cm3 SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 18. r = 7.5 mm and h = 15.2 mm. SOLUTION:

r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches

Now you can find the volume. below the top?

SOLUTION:

The planter is to be filled inches below the top, so

ANSWER: 3 823.0 in

ANSWER: 19. 3 2740.5 in SOLUTION: If two solids have the same height h and the same 21. SHIPPING A box 18 centimeters by 9 centimeters cross-sectional area B at every level, then they have by 15 centimeters is being used to ship two the same volume. So, the volume of a right prism and cylindrical candles. Each candle has a diameter of 9 an oblique one of the same height and cross sectional centimeters and a height of 15 centimeters, as shown area are same. at the right. What is the volume of the empty space in the box? r = 7.5 mm and h = 15.2 mm.

SOLUTION: The volume of the empty space is the difference of ANSWER: volumes of the rectangular prism and the cylinders. 3 2686.1 mm

20. PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil

in the planter if the planter is filled to inches

below the top? ANSWER: SOLUTION: 521.5 cm3

The planter is to be filled inches below the top, so 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that

are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3 2740.5 in

21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box? Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER: 2.52 ft Find the volume of the solid formed by each net. SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

23. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a ANSWER: triangular prism.

521.5 cm3 One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height 22. SANDCASTLES In a sandcastle competition, of the base. contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. The height of the prism is 20 cm. What should the diameter of the cylinders be?

SOLUTION: The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism. 3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3934.9 cm3

Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER: 2.52 ft Find the volume of the solid formed by each net. 24. SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. 23. However, since the middle piece is a parallelogram, it is oblique. SOLUTION: The middle piece of the net is the front of the solid. The radius is 1.8 m, the height is 4.8 m, and the slant The top and bottom pieces are the bases and the height is 6 m. pieces on the ends are the side faces. This is a triangular prism. If two solids have the same height h and the same cross-sectional area B at every level, then they have One leg of the base 14 cm and the hypotenuse 31.4 the same volume. So, the volume of a right prism and cm. Use the Pythagorean Theorem to find the height an oblique one of the same height and cross sectional of the base. area are same.

The height of the prism is 20 cm.

The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism.

ANSWER: 3 48.9 m

25. FOOD A cylindrical can of baked potato chips has a ANSWER: height of 27 centimeters and a radius of 4 3 centimeters. A new can is advertised as being 30% 3934.9 cm larger than the regular can. If both cans have the same radius, what is the height of the larger can?

SOLUTION: The volume of the smaller can is 24. The volume of the new can is 130% of the smaller SOLUTION: can, with the same radius. The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

If two solids have the same height h and the same cross-sectional area B at every level, then they have The height of the new can will be 35.1 cm. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 35.1 cm

25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can? a. When the height is tripled, h = 3h.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller When the height is tripled, the volume is multiplied by can, with the same radius. 3.

b. When the radius is tripled, r = 3r.

The height of the new can will be 35.1 cm. So, when the radius is tripled, the volume is multiplied by 9. ANSWER: 35.1 cm c. When the height and the radius are tripled, r = 3r and h = 3h. 26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION: When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

a. When the height is tripled, h = 3h.

Compare to the original volume.

The volume is multiplied by . When the height is tripled, the volume is multiplied by 3. ANSWER: b. When the radius is tripled, r = 3r. a. The volume is multiplied by 3.

b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific So, when the radius is tripled, the volume is multiplied plant will grow in it. The density by 9. of the soil sample is the ratio of its weight to its volume. c. When the height and the radius are tripled, r = 3r and h = 3h. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds?

SOLUTION: When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

Compare to the original volume.

The weight of the soil is the weight of the pot with The volume is multiplied by . soil minus the weight of the pot. W = 20 – 5 = 15 lbs. ANSWER: a. The volume is multiplied by 3. The soil density is thus: b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant 27. SOIL A soil scientist wants to determine the bulk should grow fairly well. density of a potting soil to assess how well a specific plant will grow in it. The density c. of the soil sample is the ratio of its weight to its volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound ANSWER: per square inch is desirable for root growth? Explain. 3 c. If a bag of this soil holds 2.5 cubic feet, what is its a. 0.0019 lb / in weight in pounds? b. The plant should grow well in this soil since the SOLUTION: bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round

to the nearest tenth if necessary. a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

28. SOLUTION: The solid is a combination of two rectangular prisms. The weight of the soil is the weight of the pot with The base of one rectangular prism is 5 cm by 3 cm soil minus the weight of the pot. and the height is 11 cm. The base of the other prism W = 20 – 5 = 15 lbs. is 4 cm by 3 cm and the height is 5 cm.

The soil density is thus:

3 3 b. 0.0018 lb/in is close to 0.0019 lb/in so the plant ANSWER: should grow fairly well. 3 225 cm c.

ANSWER: 29. 3 a. 0.0019 lb / in SOLUTION:

b. The plant should grow well in this soil since the The solid is a combination of a rectangular prism and 3 a right triangular prism. The total volume of the solid bulk density of 0.0019 lb / in is close to the desired is the sum of the volumes of the two rectangular 3 bulk density of 0.0018 lb / in . prisms.

c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

ANSWER: 28. 3 120 m SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

30. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

ANSWER: 225 cm3

ANSWER: 260.5 in3 29. 31. MANUFACTURING A can 12 centimeters tall fits SOLUTION: into a rubberized cylindrical holder that is 11.5 The solid is a combination of a rectangular prism and centimeters tall, including 1 centimeter for the a right triangular prism. The total volume of the solid thickness of the base of the holder. The thickness of is the sum of the volumes of the two rectangular the rim of the holder is 1 centimeter. What is the prisms. volume of the rubberized material that makes up the holder?

SOLUTION: ANSWER: The volume of the rubberized material is the 3 120 m difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 30. volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic ANSWER: centimeters. The diameter of the can is 9 260.5 in3 centimeters. What is the height? SOLUTION: 31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

ANSWER: 5.7 cm

SOLUTION: 33. A cylinder has a surface area of 144π square inches The volume of the rubberized material is the and a height of 6 inches. What is the volume? difference between the volumes of the container and the space used for the can. The container has a SOLUTION: Use the surface area formula to solve for r. radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm Find each measure to the nearest tenth. The radius is 6. Find the volume. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? ANSWER: SOLUTION: 5.7 cm Use the surface area formula to find the length of the base of the prism.

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

Find the volume.

ANSWER: 3 576 in

35. ARCHITECTURE A cylindrical stainless steel The radius is 6. Find the volume. column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter.

SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is ANSWER: when the radii are 47.5 cm and 15 cm respectively. 678.6 in3 Find the difference between the volumes. 34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 Find the volume. feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

ANSWER: SOLUTION: 3 576 in The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the 35. ARCHITECTURE A cylindrical stainless steel rectangular prism is 6 ft by 10 ft and the height is 15 column is used to hide a ventilation system in a new ft. The bases of the trapezoidal prism are 6 ft and 3 building. According to the specifications, the diameter ft long and the height of the base is 10 ft. The height of the column can be between 30 centimeters and 95 of the trapezoidal prism is 15 ft. The total volume of centimeters. The height is to be 500 centimeters. the solid is the sum of the volumes of the two What is the difference in volume between the largest prisms. and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes. ANSWER: 3 1575 ft

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume SOLUTION: of water it takes to fill the pool. Find the volume of the original container.

SOLUTION: The volume of the new container is 125% of the original container, with the same base dimensions. The swimming pool is a combination of a rectangular Use 1.25V and B to find h. prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms.

ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each ANSWER: drawing. (1 cup ≈ 14.4375 in3) 3 1575 ft SOLUTION: Sample answers: 37. CHANGING DIMENSIONS A soy milk company

is planning a promotion in which the volume of soy For any cylindrical container, we have the following milk in each container will be increased by 25%. The equation for volume: company wants the base of the container to stay the same. What will be the height of the new containers?

The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled SOLUTION: to get the desired volume. First, choose a suitable Find the volume of the original container. radius, say 1.85 in, and solve for the height.

The volume of the new container is 125% of the original container, with the same base dimensions.

Use 1.25V and B to find h. If we choose a height of say 4 in., then we can solve for the radius.

ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds

1 cup. Be sure to include the dimensions in each For any rectangular container, the volume equation is: drawing. (1 cup ≈ 14.4375 in3) SOLUTION:

Sample answers: Choose numbers for any two of the dimensions and

we can solve for the third. Let l = 2.25 in. and w = For any cylindrical container, we have the following 2.5 in. equation for volume:

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

SOLUTION: The base of the prism can be divided into 5 congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height ANSWER: 10 cm. Find the base area of each triangular prism. Sample answers:

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd 39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe Find the volume. the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50. ANSWER: 3 1100 cm ; Each triangular prism has a base area of 41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique 2 or 22 cm and a height of 10 cm. cylinders . a. GEOMETRIC Draw a right cylinder and an 40. PATIOS Mr. Thomas is planning to remove an old oblique cylinder with a height of 10 meters and a patio and install a new rectangular concrete patio 20 diameter of 6 meters. feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio b. VERBAL A square prism has a height of 10 and $700 for removal of the old patio. Which is the meters and a base edge of 6 meters. Is its volume less expensive option? Explain. greater than, less than, or equal to the volume of the cylinder? Explain. SOLUTION: Convert all of the dimensions to yards. c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the 20 feet = yd height by x or multiplying the radius by x. Explain. 12 feet = 4 yd SOLUTION: 4 in. = yd a. The oblique cylinder should look like the right cylinder (same height and size), except that it is Find the volume. pushed a little to the side, like a slinky.

b. Find the volume of each.

The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option. ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this The volume of the square prism is greater. problem, you will investigate right and oblique c. Do each scenario. cylinders . a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the Assuming x > 1, multiplying the radius by x makes 2 height by x or multiplying the radius by x. Explain. the volume x times greater. SOLUTION: 2 a. The oblique cylinder should look like the right For example, if x = 0.5, then x = 0.25, which is less cylinder (same height and size), except that it is than x. pushed a little to the side, like a slinky. ANSWER: a.

b. Find the volume of each.

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, The volume of the square prism is greater. assuming x > 1. c. Do each scenario. 42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning.

SOLUTION: Francisco; Valerie incorrectly used as the Assuming x > 1, multiplying the radius by x makes length of one side of the triangular base. Francisco 2 the volume x times greater. used a different approach, but his solution is correct.

2 Francisco used the standard formula for the volume For example, if x = 0.5, then x = 0.25, which is less of a solid, V = Bh. The area of the base, B, is one-

than x. half the apothem multiplied by the perimeter of the ANSWER: base.

ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct. b. Greater than; a square with a side length of 6 m 2 43. CHALLENGE A cylindrical can is used to fill a has an area of 36 m . A circle with a diameter of 6 container with liquid. It takes three full cans to fill the m has an area of 9π or 28.3 m2. Since the heights container. Describe possible dimensions of the are the same, the volume of the square prism is container if it is each of the following shapes. greater. a. rectangular prism c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x b. square prism makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, c. triangular prism with a right triangle as the base assuming x > 1.

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the

container is 60π in 3. SOLUTION: Francisco; Valerie incorrectly used as the a. Choose some basic values for 2 of the sides, and length of one side of the triangular base. Francisco then determine the third side. Base: 3 by 5. used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

3 by 5 by 4π

b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the

container if it is each of the following shapes.

a. rectangular prism 5 by 5 by

b. square prism c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. c. triangular prism with a right triangle as the base

SOLUTION:

3 by 4 by 10π 3 The volume of the can is 20π in . It takes three full ANSWER: cans to fill the container, so the volume of the Sample answers: container is 60π in 3. a. 3 by 5 by 4π a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet 3 by 5 by 4π gardening forum.

I am new to gardening. The nursery will deliver a b. Choose some basic values for 2 of the sides, and truckload of soil, which they say is 4 yards. I

then determine the third side. Base: 5 by 5. know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard, 5 by 5 by 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine c. Choose some basic values for 2 of the sides, and the number of cubic yards of soil needed. then determine the third side. Base: Legs: 3 by 4. 45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest values to choose.

3 by 4 by 10π Sample answer:

ANSWER: Sample answers:

a. 3 by 5 by 4π

ANSWER: b. 5 by 5 by Sample answer:

c. base with legs measuring 3 in. and 4 in., height 10π in.

SOLUTION: True; if two cylinders have the same height (h1 = h2) Sample answer: The nursery means a cubic yard, and the same lateral area (L1 = L2), the circular 3 which is 3 or 27 cubic feet. Find the volume of your bases must have the same area. garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a The radii must also be equal. volume of 50 cubic centimeters. ANSWER: SOLUTION: True; if two cylinders have the same height and the Choose 3 values that multiply to make 50. The same lateral area, the circular bases must have the 2 factors of 50 are 2, 5, 5, so these are the simplest same area. Therefore, πr h is the same for each values to choose. cylinder.

Sample answer: 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, ANSWER: so the expression representing the area varies, depending on the type of polygon it is. The base of a Sample answer: cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 46. REASONING Determine whether the following 2 statement is true or false. Explain. base of a cylinder is a circle, so its area is πr . Two cylinders with the same height and the same 48. The volume of a triangular prism is 1380 cubic

lateral area must have the same volume. centimeters. Its base is a right triangle with legs SOLUTION: measuring 8 centimeters and 15 centimeters. What is the height of the prism? True; if two cylinders have the same height (h = h ) 1 2 A 34.5 cm and the same lateral area (L1 = L2), the circular B 23 cm bases must have the same area. C 17 cm D 11.5 cm SOLUTION:

The radii must also be equal.

ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 same area. Therefore, πr h is the same for each ANSWER: cylinder. B

47. WRITING IN MATH How are the formulas for 49. A cylindrical tank used for oil storage has a height the volume of a prism and the volume of a cylinder that is half the length of its radius. If the volume of similar? How are they different? the tank is 1,122,360 ft3, what is the tank’s radius? SOLUTION: F 89.4 ft Both formulas involve multiplying the area of the G 178.8 ft base by the height. The base of a prism is a polygon, H 280.9 ft so the expression representing the area varies, J 561.8 ft depending on the type of polygon it is. The base of a SOLUTION: cylinder is a circle, so its area is πr2.

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER: B ANSWER: 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of the tank is 1,122,360 ft3, what is the tank’s radius? 51. SAT/ACT A county proposes to enact a new 0.5% F 89.4 ft property tax. What would be the additional tax G 178.8 ft amount for a landowner whose property has a H 280.9 ft taxable value of $85,000? J 561.8 ft A $4.25 SOLUTION: B $170 C $425 D $4250 E $42,500 SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C

Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if ANSWER: necessary. F

50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square? 52. SOLUTION:

be written as shown. The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. ANSWER:

Therefore, the correct choice is C.

ANSWER: Find the perimeter and area of the equilateral C triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

The perimeter is P = 3 × 10 or 30 feet.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

Find the perimeter and area of the equilateral So, the lateral area of the pyramid is about 212.1 ft2. triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER:

212.1 ft2; 255.4 ft2

The perimeter is P = 3 × 10 or 30 feet. 53.

SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular pyramid is , where So, the lateral area of the pyramid is about 212.1 ft2. L is the lateral area and B is the area of the base.

Therefore, the surface area of the pyramid is 175 Therefore, the surface area of the pyramid is about cm2. 2 255.4 ft . ANSWER: 2 2 ANSWER: 126 cm ; 175 cm 212.1 ft2; 255.4 ft2

54. SOLUTION: 53. The pyramid has a slant height of 15 inches and the SOLUTION: base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the The lateral area L of a regular pyramid is , angle formed in the triangle below is 30°. where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the lateral area of the pyramid is 126 cm2. Use a trigonometric ratio to find the measure of the apothem a.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

Find the lateral area and surface area of the pyramid.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: So, the lateral area of the pyramid is 472.5 in2. 2 2 126 cm ; 175 cm

54. SOLUTION:

The pyramid has a slant height of 15 inches and the Therefore, the surface area of the pyramid is about base is a hexagon with sides of 10.5 inches. 2 A central angle of the hexagon is or 60°, so the 758.9 in .

angle formed in the triangle below is 30°. ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

Use a trigonometric ratio to find the measure of the apothem a. SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: Find the lateral area and surface area of the 205 in2 pyramid. Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter.

SOLUTION: So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is about The diameter of the circle is about 8.3 m. 758.9 in2. ANSWER: 8.3 m ANSWER: 57. Find the diameter of a circle with an area of 102 2 2 472.5 in ; 758.9 in square centimeters.

55. BAKING Many baking pans are given a special SOLUTION: nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. The diameter of the circle is about 11.4 m. Therefore, the area that needs to be coated is 2(13 + 2 ANSWER: 9)(2) + 13(9) = 205 in . 11.4 cm ANSWER: 58. The area of a circle is 191 square feet. Find the 2 205 in radius. Find the indicated measure. Round to the SOLUTION: nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

ANSWER: 7.8 ft

59. Find the radius of a circle with an area of 271 square The diameter of the circle is about 8.3 m. inches. ANSWER: SOLUTION: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

ANSWER: 9.3 in.

Find the area of each trapezoid, rhombus, or kite. The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the

radius. 60. SOLUTION: SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 7.8 ft ANSWER: 2 59. Find the radius of a circle with an area of 271 square 120 in inches. SOLUTION:

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite. ANSWER: 2 378 m

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

Find the volume of each prism.

1. ANSWER: 3 SOLUTION: 396 in The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 3. the oblique rectangular prism shown at the right

SOLUTION:

3 If two solids have the same height h and the same The volume is 108 cm . cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 108 cm3

ANSWER: 2. 26.95 m3 SOLUTION: The volume V of a prism is V = Bh, where B is the 4. an oblique pentagonal prism with a base area of 42 area of a base and h is the height of the prism. square centimeters and a height of 5.2 centimeters

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have 5. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional SOLUTION: area are same.

ANSWER: ANSWER: 206.4 ft3 26.95 m3

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: 6. If two solids have the same height h and the same cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right prism and If two solids have the same height h and the same an oblique one of the same height and cross sectional cross-sectional area B at every level, then they have area are same. the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 218.4 cm ANSWER: Find the volume of each cylinder. Round to the 3 nearest tenth. 1357.2 m 7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION: 5. SOLUTION:

ANSWER: 1025.4 cm3

ANSWER: 8. a cylinder with a radius of 4.2 inches and a height of 3 7.4 inches 206.4 ft SOLUTION:

6. SOLUTION: If two solids have the same height h and the same ANSWER: 3 cross-sectional area B at every level, then they have 410.1 in the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross 9. MULTIPLE CHOICE A rectangular lap pool sectional area are same. measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000 D 48,000 ANSWER: 3 SOLUTION: 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters

SOLUTION: Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: D ANSWER: 1025.4 cm3 CCSS SENSE-MAKING Find the volume of each prism. 8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION: 10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

ANSWER: 3 410.1 in

9. MULTIPLE CHOICE A rectangular lap pool ANSWER: measures 80 feet long by 20 feet wide. If it needs to 3 be filled to four feet deep and each cubic foot holds 30 in 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000 D 48,000 11. SOLUTION: SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the prism is 14 m.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. ANSWER: 12-4ANSWER: Volumes of Prisms and Cylinders 3 D 539 m CCSS SENSE-MAKING Find the volume of each prism.

10. 12. SOLUTION: SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The base is a right triangle with a leg length of 9 cm

The height of the prism is 5 in. and the hypotenuse of length 15 cm.

Use the Pythagorean Theorem to find the height of the base.

ANSWER: 30 in3

The height of the prism is 6 cm.

11. SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the ANSWER: prism is 14 m. 324 cm3

ANSWER:

3 13. 539 m SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 12.

SOLUTION: The volume V of a prism is V = Bh, where B is the The base is a right triangle with a leg length of 9 cm area of a base and h is the height of the prism. and the hypotenuse of length 15 cm. 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is Use the Pythagorean Theorem to find the height of the base. eSolutions Manual - Powered by Cognero ANSWER: Page 3 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have The height of the prism is 6 cm. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 324 cm ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches 13. SOLUTION: SOLUTION: If two solids have the same height h and the same If two solids have the same height h and the same cross-sectional area B at every level, then they have cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional an oblique one of the same height and cross sectional area are same. area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is ANSWER: 1534.25 in3 ANSWER: 58.14 ft3 CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. 14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: 16. If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and r = 5 yd and h = 18 yd an oblique one of the same height and cross sectional area are same.

ANSWER: 3 1413.7 yd ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a

height of 17 inches 17. SOLUTION: SOLUTION: If two solids have the same height h and the same r = 6 cm and h = 3.6 cm. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 407.2 cm3 ANSWER: 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. 18. SOLUTION: r = 5.5 in.

16. Use the Pythagorean Theorem to find the height of SOLUTION: the cylinder. r = 5 yd and h = 18 yd

ANSWER: 3 1413.7 yd

Now you can find the volume.

17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 823.0 in

ANSWER: 407.2 cm3 19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have 18. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional SOLUTION: area are same. r = 5.5 in. r = 7.5 mm and h = 15.2 mm. Use the Pythagorean Theorem to find the height of the cylinder.

ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a

rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil Now you can find the volume. in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

ANSWER: 3 823.0 in

19.

SOLUTION: ANSWER: If two solids have the same height h and the same 3 2740.5 in cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 21. SHIPPING A box 18 centimeters by 9 centimeters an oblique one of the same height and cross sectional by 15 centimeters is being used to ship two area are same. cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown r = 7.5 mm and h = 15.2 mm. at the right. What is the volume of the empty space in the box?

ANSWER: 2686.1 mm3 SOLUTION: The volume of the empty space is the difference of 20. PLANTER A planter is in the shape of a volumes of the rectangular prism and the cylinders.

rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches

below the top? SOLUTION:

The planter is to be filled inches below the top, so ANSWER: 521.5 cm3

SOLUTION: ANSWER: 3 3 2740.5 in V = 10 ft and h = 2 ft Use the formula to find r.

Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER: 2.52 ft SOLUTION: Find the volume of the solid formed by each The volume of the empty space is the difference of net. volumes of the rectangular prism and the cylinders.

23. SOLUTION: ANSWER: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the 3 521.5 cm pieces on the ends are the side faces. This is a triangular prism. 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, One leg of the base 14 cm and the hypotenuse 31.4 and 10 cubic feet of sand. To transport the correct cm. Use the Pythagorean Theorem to find the height amount of sand, they want to create cylinders that of the base. are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION: The height of the prism is 20 cm.

3 V = 10 ft and h = 2 ft Use the formula to find r. The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism.

ANSWER: Therefore, the diameter of the cylinders should be 3934.9 cm3 about 2.52 ft.

ANSWER: 2.52 ft Find the volume of the solid formed by each net.

24. SOLUTION: The circular bases at the top and bottom of the net 23. indicate that this is a cylinder. If the middle piece SOLUTION: were a rectangle, then the prism would be right. The middle piece of the net is the front of the solid. However, since the middle piece is a parallelogram, it The top and bottom pieces are the bases and the is oblique. pieces on the ends are the side faces. This is a triangular prism. The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m. One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height If two solids have the same height h and the same of the base. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. The height of the prism is 20 cm.

The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism.

ANSWER: 3 ANSWER: 48.9 m 3 3934.9 cm 25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

24. SOLUTION: SOLUTION: The volume of the smaller can is The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece The volume of the new can is 130% of the smaller were a rectangle, then the prism would be right. can, with the same radius. However, since the middle piece is a parallelogram, it is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. ANSWER: b. The radius is tripled. 3 48.9 m c. Both the radius and the height are tripled. d. The dimensions are exchanged. 25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 SOLUTION: centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

a. When the height is tripled, h = 3h.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm So, when the radius is tripled, the volume is multiplied by 9. 26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. c. When the height and the radius are tripled, r = 3r Describe how each change affects the volume of the and h = 3h. cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged.

SOLUTION:

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h a. = 5 cm. When the height is tripled, h = 3h.

Compare to the original volume.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r. The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

So, when the radius is tripled, the volume is multiplied SOIL by 9. 27. A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density c. When the height and the radius are tripled, r = 3r of the soil sample is the ratio of its weight to its and h = 3h. volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its When the height and the radius are tripled, the weight in pounds? volume is multiplied by 27. SOLUTION: d. When the dimensions are exchanged, r = 8 and h = 5 cm.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

Compare to the original volume.

The volume is multiplied by .

ANSWER: The weight of the soil is the weight of the pot with soil minus the weight of the pot. a. The volume is multiplied by 3. W = 20 – 5 = 15 lbs. 2 b. The volume is multiplied by 3 or 9. 3 c. The volume is multiplied by 3 or 27. The soil density is thus:

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific 3 3 b. 0.0018 lb/in is close to 0.0019 lb/in so the plant plant will grow in it. The density should grow fairly well. of the soil sample is the ratio of its weight to its

volume. c.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? ANSWER: 3 SOLUTION: a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb a. First calculate the volume of soil in the pot. Then Find the volume of each composite solid. Round divide the weight of the soil by the volume. to the nearest tenth if necessary.

28. The weight of the soil is the weight of the pot with soil minus the weight of the pot. SOLUTION: W = 20 – 5 = 15 lbs. The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm The soil density is thus: and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

c. ANSWER: 225 cm3

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the 29. bulk density of 0.0019 lb / in3 is close to the desired SOLUTION: 3 bulk density of 0.0018 lb / in . The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid c. 8.3 lb is the sum of the volumes of the two rectangular prisms. Find the volume of each composite solid. Round to the nearest tenth if necessary.

28.

SOLUTION: ANSWER: The solid is a combination of two rectangular prisms. 3 The base of one rectangular prism is 5 cm by 3 cm 120 m and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

30. SOLUTION: The solid is a combination of a rectangular prism and ANSWER: two half cylinders. 225 cm3

29. ANSWER: SOLUTION: 260.5 in3 The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid 31. MANUFACTURING A can 12 centimeters tall fits is the sum of the volumes of the two rectangular into a rubberized cylindrical holder that is 11.5 prisms. centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

ANSWER:

3 120 m SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. 30. The empty space used to keep the can has a radius SOLUTION: of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 The solid is a combination of a rectangular prism and volume V of a cylinder is V = πr h, where r is the two half cylinders. radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is 3 about 304.1 cm .

ANSWER:

3 304.1 cm ANSWER: 3 Find each measure to the nearest tenth. 260.5 in 32. A cylindrical can has a volume of 363 cubic 31. MANUFACTURING A can 12 centimeters tall fits centimeters. The diameter of the can is 9

into a rubberized cylindrical holder that is 11.5 centimeters. What is the height? centimeters tall, including 1 centimeter for the SOLUTION: thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

SOLUTION: ANSWER: The volume of the rubberized material is the 5.7 cm difference between the volumes of the container and the space used for the can. The container has a 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? radius of and a height of 11.5 cm. SOLUTION: The empty space used to keep the can has a radius Use the surface area formula to solve for r.

of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 The radius is 6. Find the volume.

centimeters. What is the height? SOLUTION:

ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432 ANSWER: square inches, a height of 6 inches, and a width of 12 5.7 cm inches. What is the volume? SOLUTION: 33. A cylinder has a surface area of 144π square inches Use the surface area formula to find the length of the and a height of 6 inches. What is the volume? base of the prism. SOLUTION: Use the surface area formula to solve for r.

Find the volume.

ANSWER:

3 The radius is 6. Find the volume. 576 in

ANSWER: SOLUTION: 3 The volume will be the highest when the diameter is 678.6 in 95 cm and will be the lowest when it is 30 cm.That is

34. A rectangular prism has a surface area of 432 when the radii are 47.5 cm and 15 cm respectively.

square inches, a height of 6 inches, and a width of 12

inches. What is the volume? Find the difference between the volumes.

SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER:

3,190,680.0 cm3 Find the volume. 36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

ANSWER: 3 576 in SOLUTION: 35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new The swimming pool is a combination of a rectangular building. According to the specifications, the diameter prism and a trapezoidal prism. The base of the of the column can be between 30 centimeters and 95 rectangular prism is 6 ft by 10 ft and the height is 15 centimeters. The height is to be 500 centimeters. ft. The bases of the trapezoidal prism are 6 ft and 3 What is the difference in volume between the largest ft long and the height of the base is 10 ft. The height and smallest possible column? Round to the nearest of the trapezoidal prism is 15 ft. The total volume of tenth cubic centimeter. the solid is the sum of the volumes of the two prisms. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 3 1575 ft

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool. SOLUTION: Find the volume of the original container.

SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the The volume of the new container is 125% of the rectangular prism is 6 ft by 10 ft and the height is 15 original container, with the same base dimensions. ft. The bases of the trapezoidal prism are 6 ft and 3 Use 1.25V and B to find h. ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms.

ANSWER:

ANSWER: 38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 3 1575 ft 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) 37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy SOLUTION: milk in each container will be increased by 25%. The Sample answers: company wants the base of the container to stay the same. What will be the height of the new containers? For any cylindrical container, we have the following equation for volume:

SOLUTION: Find the volume of the original container. The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each 3 drawing. (1 cup ≈ 14.4375 in )

SOLUTION: For any rectangular container, the volume equation is: Sample answers:

For any cylindrical container, we have the following equation for volume: Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER: For any rectangular container, the volume equation is: Sample answers:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

SOLUTION: The base of the prism can be divided into 5 ANSWER: congruent triangles of a base 8 cm and the Sample answers: corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 39. Find the volume of the regular pentagonal prism by 12 feet = 4 yd dividing it into five equal triangular prisms. Describe 4 in. = yd the base area and height of each triangular prism. Find the volume.

Therefore, the volume of the pentagonal prism is Therefore, the second contractor is a less expensive option. ANSWER: ANSWER: 3 3 Because 2.96 yd of concrete are needed, the 1100 cm ; Each triangular prism has a base area of second contractor is less expensive at $2181.50. 2 or 22 cm and a height of 10 cm. 41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique 40. PATIOS Mr. Thomas is planning to remove an old cylinders patio and install a new rectangular concrete patio 20 . feet long, 12 feet wide, and 4 inches thick. One a. GEOMETRIC Draw a right cylinder and an contractor bid $2225 for the project. A second oblique cylinder with a height of 10 meters and a contractor bid $500 per cubic yard for the new patio diameter of 6 meters. and $700 for removal of the old patio. Which is the less expensive option? Explain. b. VERBAL A square prism has a height of 10 SOLUTION: meters and a base edge of 6 meters. Is its volume Convert all of the dimensions to yards. greater than, less than, or equal to the volume of the cylinder? Explain.

20 feet = yd c. ANALYTICAL Describe which change affects

12 feet = 4 yd the volume of the cylinder more: multiplying the 4 in. = yd height by x or multiplying the radius by x. Explain.

SOLUTION: Find the volume. a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Find the volume of each. The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique

cylinders The volume of the square prism is greater. . c. Do each scenario. a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain.

SOLUTION: Assuming x > 1, multiplying the radius by x makes 2 a. The oblique cylinder should look like the right the volume x times greater. cylinder (same height and size), except that it is pushed a little to the side, like a slinky. 2 For example, if x = 0.5, then x = 0.25, which is less than x.

ANSWER: a.

b. Find the volume of each.

c. Multiplying the radius by x ; since the volume is

The volume of the square prism is greater. represented by π r 2 h, multiplying the height by x c. Do each scenario. makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

Assuming x > 1, multiplying the radius by x makes SOLUTION: 2 the volume x times greater. Francisco; Valerie incorrectly used as the

2 length of one side of the triangular base. Francisco For example, if x = 0.5, then x = 0.25, which is less used a different approach, but his solution is correct. than x. Francisco used the standard formula for the volume ANSWER: of a solid, V = Bh. The area of the base, B, is one- a. half the apothem multiplied by the perimeter of the base.

ANSWER:

Francisco; Valerie incorrectly used as the b. Greater than; a square with a side length of 6 m 2 length of one side of the triangular base. Francisco has an area of 36 m . A circle with a diameter of 6 used a different approach, but his solution is correct. m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is 43. CHALLENGE A cylindrical can is used to fill a greater. container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the

c. Multiplying the radius by x ; since the volume is container if it is each of the following shapes.

represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the a. rectangular prism

radius by x makes the volume x 2 times greater, b. assuming x > 1. square prism

42. CCSS CRITIQUE Franciso and Valerie each c. triangular prism with a right triangle as the base calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning.

SOLUTION:

in3 SOLUTION: The volume of the can is 20π . It takes three full cans to fill the container, so the volume of the Francisco; Valerie incorrectly used as the container is 60π in 3. length of one side of the triangular base. Francisco used a different approach, but his solution is correct. a. Choose some basic values for 2 of the sides, and

Francisco used the standard formula for the volume then determine the third side. Base: 3 by 5. of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

3 by 5 by 4π ANSWER: Francisco; Valerie incorrectly used as the b. Choose some basic values for 2 of the sides, and

length of one side of the triangular base. Francisco then determine the third side. Base: 5 by 5. used a different approach, but his solution is correct.

a. rectangular prism

b. square prism 5 by 5 by

c. triangular prism with a right triangle as the base c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4.

SOLUTION:

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the 3 by 4 by 10π 3 container is 60π in . ANSWER:

Sample answers: a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. a. 3 by 5 by 4π

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height π 10 in.

3 by 5 by 4π 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet b. Choose some basic values for 2 of the sides, and gardening forum. then determine the third side. Base: 5 by 5. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. 5 by 5 by ANSWER: c. Choose some basic values for 2 of the sides, and Sample answer: The nursery means a cubic yard, then determine the third side. Base: Legs: 3 by 4. which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest 3 by 4 by 10π values to choose. ANSWER: Sample answer: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height ANSWER: 10π in. Sample answer:

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum.

SOLUTION: lateral area must have the same volume. Sample answer: The nursery means a cubic yard, SOLUTION: 3 which is 3 or 27 cubic feet. Find the volume of your True; if two cylinders have the same height (h1 = h2) garden in cubic feet and divide by 27 to determine and the same lateral area (L = L ), the circular the number of cubic yards of soil needed. 1 2 bases must have the same area. ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters.

SOLUTION: The radii must also be equal. Choose 3 values that multiply to make 50. The ANSWER: factors of 50 are 2, 5, 5, so these are the simplest values to choose. True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 Sample answer: same area. Therefore, πr h is the same for each cylinder.

47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different?

SOLUTION: ANSWER: Both formulas involve multiplying the area of the Sample answer: base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the 46. REASONING Determine whether the following area of the base by the height. The base of a prism is statement is true or false. Explain. a polygon, so the expression representing the area Two cylinders with the same height and the same varies, depending on the type of polygon it is. The 2 lateral area must have the same volume. base of a cylinder is a circle, so its area is πr . SOLUTION: 48. The volume of a triangular prism is 1380 cubic True; if two cylinders have the same height (h1 = h2) centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is and the same lateral area (L1 = L2), the circular the height of the prism? bases must have the same area. A 34.5 cm B 23 cm C 17 cm D 11.5 cm SOLUTION:

The radii must also be equal.

ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 same area. Therefore, πr h is the same for each cylinder.

47. WRITING IN MATH How are the formulas for ANSWER: the volume of a prism and the volume of a cylinder B similar? How are they different? 49. A cylindrical tank used for oil storage has a height SOLUTION: that is half the length of its radius. If the volume of Both formulas involve multiplying the area of the the tank is 1,122,360 ft3, what is the tank’s radius? base by the height. The base of a prism is a polygon, F 89.4 ft so the expression representing the area varies, G 178.8 ft depending on the type of polygon it is. The base of a H 2 280.9 ft cylinder is a circle, so its area is πr . J 561.8 ft ANSWER: SOLUTION: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr .

D 11.5 cm SOLUTION: ANSWER: F 50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square?

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can ANSWER: be written as shown. B 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of ANSWER: 3 the tank is 1,122,360 ft , what is the tank’s radius? F 89.4 ft G 178.8 ft H 280.9 ft 51. SAT/ACT A county proposes to enact a new 0.5% J 561.8 ft property tax. What would be the additional tax amount for a landowner whose property has a SOLUTION: taxable value of $85,000? A $4.25 B $170 C $425 D $4250 E $42,500 SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C ANSWER: F Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if 50. SHORT RESPONSE What is the ratio of the area necessary. of the circle to the area of the square?

52.

SOLUTION: SOLUTION: The radius of the circle is 2x and the length of each The lateral area L of a regular pyramid is , side of the square is 4x. So, the ratio of the areas can be written as shown. where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the ANSWER: congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

Therefore, the correct choice is C.

ANSWER: C

Find the lateral area and surface area of each Find the perimeter and area of the equilateral regular pyramid. Round to the nearest tenth if triangle for the base. Use the Pythagorean Theorem necessary. to find the height h of the triangle.

52. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

The perimeter is P = 3 × 10 or 30 feet.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. 2 So, the lateral area of the pyramid is about 212.1 ft .

Therefore, the surface area of the pyramid is about 2 255.4 ft . ANSWER: 212.1 ft2; 255.4 ft2

The perimeter is P = 3 × 10 or 30 feet.

53. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of 2 So, the area of the base B is ft . the base. Here, the base is a square of side 7 cm and the slant Find the lateral area L and surface area S of the height is 9 cm. regular pyramid.

So, the lateral area of the pyramid is 126 cm2.

So, the lateral area of the pyramid is about 212.1 ft2. The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

Therefore, the surface area of the pyramid is about 2 255.4 ft . Therefore, the surface area of the pyramid is 175 2 ANSWER: cm . 2 2 212.1 ft ; 255.4 ft ANSWER: 2 2 126 cm ; 175 cm

53. 54. SOLUTION: SOLUTION: The lateral area L of a regular pyramid is , The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. where is the slant height and P is the perimeter of A central angle of the hexagon is or 60°, so the the base. angle formed in the triangle below is 30°. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the lateral area of the pyramid is 126 cm2.

Use a trigonometric ratio to find the measure of the The surface area S of a regular apothem a. pyramid is , where L is the lateral area and B is the area of the base.

Find the lateral area and surface area of the pyramid. Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

So, the lateral area of the pyramid is 472.5 in2.

54. SOLUTION: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°. Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 Find the lateral area and surface area of the 9)(2) + 13(9) = 205 in . pyramid. ANSWER: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the So, the lateral area of the pyramid is 472.5 in2. diameter. SOLUTION:

Therefore, the surface area of the pyramid is about 758.9 in2. The diameter of the circle is about 8.3 m.

ANSWER: ANSWER: 2 2 8.3 m 472.5 in ; 758.9 in

55. BAKING Many baking pans are given a special 57. Find the diameter of a circle with an area of 102 nonstick coating. A rectangular cake pan is 9 inches square centimeters. by 13 inches by 2 inches deep. What is the area of SOLUTION: the inside of the pan that needs to be coated?

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in . The diameter of the circle is about 11.4 m.

ANSWER: ANSWER: 2 205 in 11.4 cm Find the indicated measure. Round to the 58. The area of a circle is 191 square feet. Find the nearest tenth. radius. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION: SOLUTION:

ANSWER: The diameter of the circle is about 8.3 m. 7.8 ft ANSWER: 59. Find the radius of a circle with an area of 271 square 8.3 m inches. SOLUTION: 57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

ANSWER:

9.3 in. The diameter of the circle is about 11.4 m. Find the area of each trapezoid, rhombus, or ANSWER: kite. 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION: 60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches. ANSWER: 2 SOLUTION: 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of

the height h and the sum of the lengths of its bases,

b1 and b2. ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

ANSWER: 2 60. 378 m SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

Find the volume of each prism.

ANSWER: 3 396 in

1. 3. the oblique rectangular prism shown at the right SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 108 cm3

ANSWER: 26.95 m3

4. an oblique pentagonal prism with a base area of 42 2. square centimeters and a height of 5.2 centimeters SOLUTION: SOLUTION: The volume V of a prism is V = Bh, where B is the If two solids have the same height h and the same area of a base and h is the height of the prism. cross-sectional area B at every level, then they have

the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: ANSWER: 3 3 396 in 218.4 cm

3. the oblique rectangular prism shown at the right Find the volume of each cylinder. Round to the nearest tenth.

5. SOLUTION: If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 206.4 ft

ANSWER: 26.95 m3

4. an oblique pentagonal prism with a base area of 42 6.

square centimeters and a height of 5.2 centimeters SOLUTION: SOLUTION: If two solids have the same height h and the same If two solids have the same height h and the same cross-sectional area B at every level, then they have cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder the same volume. So, the volume of a right prism and and an oblique one of the same height and cross an oblique one of the same height and cross sectional sectional area are same. area are same.

ANSWER: ANSWER: 3 3 1357.2 m 218.4 cm 7. a cylinder with a diameter of 16 centimeters and a Find the volume of each cylinder. Round to the height of 5.1 centimeters nearest tenth. SOLUTION:

5. SOLUTION:

ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION: ANSWER: 206.4 ft3

ANSWER: 6. 3 410.1 in SOLUTION: If two solids have the same height h and the same 9. MULTIPLE CHOICE A rectangular lap pool cross-sectional area B at every level, then they have measures 80 feet long by 20 feet wide. If it needs to the same volume. So, the volume of a right cylinder be filled to four feet deep and each cubic foot holds and an oblique one of the same height and cross 7.5 gallons, how many gallons will it take to fill the sectional area are same. lap pool? A 4000 B 6400 C 30,000 D 48,000 SOLUTION:

ANSWER: 3 1357.2 m Each cubic foot holds 7.5 gallons of water. So, the 7. a cylinder with a diameter of 16 centimeters and a amount of water required to fill the pool is 6400(7.5) height of 5.1 centimeters = 48,000. SOLUTION: Therefore, the correct choice is D.

ANSWER: D CCSS SENSE-MAKING Find the volume of each prism. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 10. 7.4 inches SOLUTION: SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

ANSWER: 3 ANSWER: 410.1 in 30 in3 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 11. B 6400 C 30,000 SOLUTION: D 48,000 The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the SOLUTION: prism is 14 m.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000. ANSWER: 3 Therefore, the correct choice is D. 539 m

ANSWER: D CCSS SENSE-MAKING Find the volume of each prism. 12. SOLUTION: The base is a right triangle with a leg length of 9 cm 10. and the hypotenuse of length 15 cm.

SOLUTION: Use the Pythagorean Theorem to find the height of The base is a rectangle of length 3 in. and width 2 in. the base.

The height of the prism is 5 in.

ANSWER: 30 in3

The height of the prism is 6 cm.

11. ANSWER: SOLUTION: 3 The base is a triangle with a base length of 11 m and 324 cm the corresponding height of 7 m. The height of the prism is 14 m.

13.

SOLUTION: ANSWER: If two solids have the same height h and the same 3 539 m cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 12. 2 SOLUTION: B = 11.4 ft and h = 5.1 ft. Therefore, the volume is The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. ANSWER: Use the Pythagorean Theorem to find the height of 3 58.14 ft the base. 14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. The height of the prism is 6 cm.

ANSWER: ANSWER: 12-4 Volumes of Prisms and Cylinders 3 324 cm3 2040 cm

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. ANSWER: 3 2 1534.25 in B = 11.4 ft and h = 5.1 ft. Therefore, the volume is CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 16. centimeters and with a base area of 136 square centimeters SOLUTION: r = 5 yd and h = 18 yd SOLUTION:

ANSWER: 3 1413.7 yd

ANSWER: 3 2040 cm 17.

15. a square prism with a base edge of 9.5 inches and a SOLUTION: height of 17 inches r = 6 cm and h = 3.6 cm.

area are same. ANSWER: eSolutions Manual - Powered by Cognero Page 4 407.2 cm3

ANSWER: 1534.25 in3 18.

CCSS SENSE-MAKING Find the volume of SOLUTION: each cylinder. Round to the nearest tenth. r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

16. SOLUTION: r = 5 yd and h = 18 yd

ANSWER: 3 1413.7 yd Now you can find the volume.

17. SOLUTION: r = 6 cm and h = 3.6 cm. ANSWER:

3 823.0 in

19. ANSWER: SOLUTION: 407.2 cm3 If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

18. r = 7.5 mm and h = 15.2 mm. SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil

in the planter if the planter is filled to inches

below the top? Now you can find the volume. SOLUTION:

The planter is to be filled inches below the top, so

ANSWER: 3 823.0 in

ANSWER: 3 19. 2740.5 in

SOLUTION: 21. SHIPPING A box 18 centimeters by 9 centimeters If two solids have the same height h and the same by 15 centimeters is being used to ship two cross-sectional area B at every level, then they have cylindrical candles. Each candle has a diameter of 9 the same volume. So, the volume of a right prism and centimeters and a height of 15 centimeters, as shown an oblique one of the same height and cross sectional at the right. What is the volume of the empty space area are same. in the box?

r = 7.5 mm and h = 15.2 mm.

SOLUTION: The volume of the empty space is the difference of ANSWER: volumes of the rectangular prism and the cylinders. 3 2686.1 mm

20. PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? ANSWER: 3 SOLUTION: 521.5 cm

amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3 2740.5 in

21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space Therefore, the diameter of the cylinders should be in the box? about 2.52 ft.

ANSWER: 2.52 ft Find the volume of the solid formed by each net.

SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

23. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism. ANSWER: 521.5 cm3 One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height 22. SANDCASTLES In a sandcastle competition, of the base. contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. The height of the prism is 20 cm. What should the diameter of the cylinders be? The volume V of a prism is V = Bh, where B is the SOLUTION: area of the base, h is the height of the prism.

3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3934.9 cm3

Therefore, the diameter of the cylinders should be about 2.52 ft.

The top and bottom pieces are the bases and the height is 6 m. pieces on the ends are the side faces. This is a triangular prism. If two solids have the same height h and the same cross-sectional area B at every level, then they have One leg of the base 14 cm and the hypotenuse 31.4 the same volume. So, the volume of a right prism and cm. Use the Pythagorean Theorem to find the height an oblique one of the same height and cross sectional

of the base. area are same.

The height of the prism is 20 cm.

The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism.

ANSWER: 3 48.9 m

25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 ANSWER: centimeters. A new can is advertised as being 30% 3934.9 cm3 larger than the regular can. If both cans have the same radius, what is the height of the larger can?

SOLUTION: The volume of the smaller can is

24. The volume of the new can is 130% of the smaller SOLUTION: can, with the same radius. The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

If two solids have the same height h and the same cross-sectional area B at every level, then they have The height of the new can will be 35.1 cm. the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional area are same. 35.1 cm

25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the

same radius, what is the height of the larger can? a. When the height is tripled, h = 3h.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller When the height is tripled, the volume is multiplied by can, with the same radius. 3. b. When the radius is tripled, r = 3r.

So, when the radius is tripled, the volume is multiplied The height of the new can will be 35.1 cm. by 9. ANSWER: c. 35.1 cm When the height and the radius are tripled, r = 3r and h = 3h. 26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. When the height and the radius are tripled, the SOLUTION: volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

a. When the height is tripled, h = 3h.

Compare to the original volume.

The volume is multiplied by . When the height is tripled, the volume is multiplied by 3. ANSWER: b. When the radius is tripled, r = 3r. a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density So, when the radius is tripled, the volume is multiplied of the soil sample is the ratio of its weight to its by 9. volume.

c. When the height and the radius are tripled, r = 3r a. If the weight of the container with the soil is 20 and h = 3h. pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION:

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

Compare to the original volume.

The weight of the soil is the weight of the pot with

The volume is multiplied by . soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

ANSWER: The soil density is thus: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. 27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific c. plant will grow in it. The density of the soil sample is the ratio of its weight to its volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound ANSWER: 3 per square inch is desirable for root growth? Explain. a. 0.0019 lb / in c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? b. The plant should grow well in this soil since the 3 SOLUTION: bulk density of 0.0019 lb / in is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

28. SOLUTION:

The solid is a combination of two rectangular prisms. The weight of the soil is the weight of the pot with The base of one rectangular prism is 5 cm by 3 cm soil minus the weight of the pot. and the height is 11 cm. The base of the other prism W = 20 – 5 = 15 lbs. is 4 cm by 3 cm and the height is 5 cm.

The soil density is thus:

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant ANSWER: should grow fairly well. 225 cm3

ANSWER: 29. 3 a. 0.0019 lb / in SOLUTION: The solid is a combination of a rectangular prism and b. The plant should grow well in this soil since the a right triangular prism. The total volume of the solid bulk density of 0.0019 lb / in3 is close to the desired is the sum of the volumes of the two rectangular 3 prisms. bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

ANSWER: 225 cm3

ANSWER: 260.5 in3

29. 31. MANUFACTURING A can 12 centimeters tall fits SOLUTION: into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the The solid is a combination of a rectangular prism and thickness of the base of the holder. The thickness of a right triangular prism. The total volume of the solid the rim of the holder is 1 centimeter. What is the is the sum of the volumes of the two rectangular volume of the rubberized material that makes up the prisms. holder?

SOLUTION: ANSWER: The volume of the rubberized material is the 3 difference between the volumes of the container and 120 m the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the 30. radius of the base and h is the height of the cylinder. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic ANSWER: centimeters. The diameter of the can is 9 centimeters. What is the height? 260.5 in3 SOLUTION: 31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

ANSWER: 5.7 cm

SOLUTION: 33. A cylinder has a surface area of 144π square inches The volume of the rubberized material is the and a height of 6 inches. What is the volume? difference between the volumes of the container and SOLUTION: the space used for the can. The container has a Use the surface area formula to solve for r. radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER:

3 304.1 cm The radius is 6. Find the volume. Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: ANSWER: Use the surface area formula to find the length of the 5.7 cm base of the prism.

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

Find the volume.

ANSWER: 3 576 in

ARCHITECTURE 35. A cylindrical stainless steel column is used to hide a ventilation system in a new The radius is 6. Find the volume. building. According to the specifications, the diameter

of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION:

The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is

ANSWER: when the radii are 47.5 cm and 15 cm respectively.

3 678.6 in Find the difference between the volumes.

ANSWER: 3,190,680.0 cm3

CCSS MODELING 36. The base of a rectangular swimming pool is sloped so one end of the pool is 6 Find the volume. feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

ANSWER: SOLUTION: 3 The swimming pool is a combination of a rectangular 576 in prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 35. ARCHITECTURE A cylindrical stainless steel ft. The bases of the trapezoidal prism are 6 ft and 3 column is used to hide a ventilation system in a new ft long and the height of the base is 10 ft. The height building. According to the specifications, the diameter of the trapezoidal prism is 15 ft. The total volume of of the column can be between 30 centimeters and 95 the solid is the sum of the volumes of the two centimeters. The height is to be 500 centimeters. prisms. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes. ANSWER: 3 1575 ft

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown SOLUTION: in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool. Find the volume of the original container.

The volume of the new container is 125% of the SOLUTION: original container, with the same base dimensions. The swimming pool is a combination of a rectangular Use 1.25V and B to find h. prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two

prisms.

ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each 3 ANSWER: drawing. (1 cup ≈ 14.4375 in ) 3 1575 ft SOLUTION: Sample answers: 37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy For any cylindrical container, we have the following milk in each container will be increased by 25%. The equation for volume: company wants the base of the container to stay the same. What will be the height of the new containers?

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h. If we choose a height of say 4 in., then we can solve for the radius.

ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each For any rectangular container, the volume equation is: drawing. (1 cup ≈ 14.4375 in3)

SOLUTION: Sample answers: Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = For any cylindrical container, we have the following 2.5 in. equation for volume:

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

SOLUTION: The base of the prism can be divided into 5 congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal

prism is a combination of 5 triangular prisms of height ANSWER: 10 cm. Find the base area of each triangular prism. Sample answers:

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

39. Find the volume of the regular pentagonal prism by Find the volume. dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

SOLUTION: The base of the prism can be divided into 5 congruent triangles of a base 8 cm and the The total cost for the second contractor is about corresponding height 5.5 cm. So, the pentagonal . prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism. Therefore, the second contractor is a less expensive option. ANSWER: Therefore, the volume of the pentagonal prism is 3 Because 2.96 yd of concrete are needed, the second contractor is less expensive at $2181.50. ANSWER: MULTIPLE REPRESENTATIONS 3 41. In this 1100 cm ; Each triangular prism has a base area of problem, you will investigate right and oblique 2 cylinders or 22 cm and a height of 10 cm. . a. GEOMETRIC Draw a right cylinder and an 40. PATIOS Mr. Thomas is planning to remove an old oblique cylinder with a height of 10 meters and a patio and install a new rectangular concrete patio 20 diameter of 6 meters. feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second b. VERBAL A square prism has a height of 10 contractor bid $500 per cubic yard for the new patio meters and a base edge of 6 meters. Is its volume and $700 for removal of the old patio. Which is the greater than, less than, or equal to the volume of less expensive option? Explain. the cylinder? Explain. SOLUTION: Convert all of the dimensions to yards. c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. 20 feet = yd 12 feet = 4 yd SOLUTION: 4 in. = yd a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky. Find the volume.

b. Find the volume of each.

The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50. The volume of the square prism is greater. 41. MULTIPLE REPRESENTATIONS In this c. Do each scenario. problem, you will investigate right and oblique cylinders . a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects Assuming x > 1, multiplying the radius by x makes the volume of the cylinder more: multiplying the 2 height by x or multiplying the radius by x. Explain. the volume x times greater.

SOLUTION: For example, if x = 0.5, then x2 = 0.25, which is less a. The oblique cylinder should look like the right than x. cylinder (same height and size), except that it is pushed a little to the side, like a slinky. ANSWER: a.

b. Find the volume of each.

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, The volume of the square prism is greater. assuming x > 1. c. Do each scenario. 42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco Assuming x > 1, multiplying the radius by x makes used a different approach, but his solution is correct. 2 the volume x times greater. Francisco used the standard formula for the volume For example, if x = 0.5, then x2 = 0.25, which is less of a solid, V = Bh. The area of the base, B, is one- than x. half the apothem multiplied by the perimeter of the base. ANSWER: a.

ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

b. Greater than; a square with a side length of 6 m 43. CHALLENGE A cylindrical can is used to fill a 2 has an area of 36 m . A circle with a diameter of 6 container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the m has an area of 9π or 28.3 m2. Since the heights container if it is each of the following shapes. are the same, the volume of the square prism is

greater. a. rectangular prism c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x b. square prism makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, c. triangular prism with a right triangle as the base assuming x > 1.

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the container is 60π in 3. SOLUTION: a. Choose some basic values for 2 of the sides, and Francisco; Valerie incorrectly used as the then determine the third side. Base: 3 by 5. length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the

base.

3 by 5 by 4π

5 by 5 by a. rectangular prism

c. Choose some basic values for 2 of the sides, and b. square prism then determine the third side. Base: Legs: 3 by 4.

c. triangular prism with a right triangle as the base

SOLUTION: 3 by 4 by 10π

3 ANSWER: The volume of the can is 20π in . It takes three full Sample answers: cans to fill the container, so the volume of the

container is 60π in 3. a. 3 by 5 by 4π

a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum. 3 by 5 by 4π I am new to gardening. The nursery will deliver a b. Choose some basic values for 2 of the sides, and truckload of soil, which they say is 4 yards. I then determine the third side. Base: 5 by 5. know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard, 3 5 by 5 by which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine c. Choose some basic values for 2 of the sides, and the number of cubic yards of soil needed. then determine the third side. Base: Legs: 3 by 4. 45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest values to choose.

Sample answer: 3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π ANSWER: b. 5 by 5 by Sample answer:

c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum. 46. REASONING Determine whether the following statement is true or false. Explain. I am new to gardening. The nursery will deliver a Two cylinders with the same height and the same truckload of soil, which they say is 4 yards. I lateral area must have the same volume. know that a yard is 3 feet, but what is a yard of SOLUTION: soil? How do I know what to order? True; if two cylinders have the same height (h1 = h2) SOLUTION: and the same lateral area (L = L ), the circular Sample answer: The nursery means a cubic yard, 1 2 3 bases must have the same area. which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. The radii must also be equal. 45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. ANSWER: True; if two cylinders have the same height and the SOLUTION: same lateral area, the circular bases must have the Choose 3 values that multiply to make 50. The 2 factors of 50 are 2, 5, 5, so these are the simplest same area. Therefore, πr h is the same for each values to choose. cylinder.

Sample answer: 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, ANSWER: depending on the type of polygon it is. The base of a Sample answer: cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area

varies, depending on the type of polygon it is. The 2 46. REASONING Determine whether the following base of a cylinder is a circle, so its area is πr . statement is true or false. Explain. Two cylinders with the same height and the same 48. The volume of a triangular prism is 1380 cubic lateral area must have the same volume. centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is SOLUTION: the height of the prism? True; if two cylinders have the same height (h1 = h2) A 34.5 cm B 23 cm and the same lateral area (L1 = L2), the circular C bases must have the same area. 17 cm D 11.5 cm SOLUTION:

The radii must also be equal.

ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 ANSWER: same area. Therefore, πr h is the same for each B cylinder. 49. A cylindrical tank used for oil storage has a height WRITING IN MATH 47. How are the formulas for that is half the length of its radius. If the volume of the volume of a prism and the volume of a cylinder 3 similar? How are they different? the tank is 1,122,360 ft , what is the tank’s radius? F 89.4 ft SOLUTION: G 178.8 ft Both formulas involve multiplying the area of the H 280.9 ft base by the height. The base of a prism is a polygon, J 561.8 ft so the expression representing the area varies, depending on the type of polygon it is. The base of a SOLUTION: cylinder is a circle, so its area is πr2.

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER: B ANSWER:

49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of 3 the tank is 1,122,360 ft , what is the tank’s radius? 51. SAT/ACT A county proposes to enact a new 0.5% F 89.4 ft property tax. What would be the additional tax G 178.8 ft amount for a landowner whose property has a H 280.9 ft taxable value of $85,000? J 561.8 ft A $4.25 B $170 SOLUTION: C $425 D $4250 E $42,500 SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary. ANSWER: F

50. SHORT RESPONSE What is the ratio of the area

of the circle to the area of the square? 52. SOLUTION:

The lateral area L of a regular pyramid is , SOLUTION: where is the slant height and P is the perimeter of The radius of the circle is 2x and the length of each the base. side of the square is 4x. So, the ratio of the areas can be written as shown. The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

ANSWER:

Therefore, the correct choice is C. ANSWER: Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem C to find the height h of the triangle. Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. The perimeter is P = 3 × 10 or 30 feet.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2. Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 2 2 212.1 ft ; 255.4 ft

The perimeter is P = 3 × 10 or 30 feet. 53. SOLUTION:

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular pyramid is , where So, the lateral area of the pyramid is about 212.1 ft2. L is the lateral area and B is the area of the base.

Therefore, the surface area of the pyramid is 175

2 Therefore, the surface area of the pyramid is about cm . 2 255.4 ft . ANSWER: 2 2 ANSWER: 126 cm ; 175 cm 212.1 ft2; 255.4 ft2

54. SOLUTION: 53. The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. SOLUTION: A central angle of the hexagon is or 60°, so the The lateral area L of a regular pyramid is , angle formed in the triangle below is 30°. where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

2 Use a trigonometric ratio to find the measure of the So, the lateral area of the pyramid is 126 cm . apothem a.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

Find the lateral area and surface area of the pyramid.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: So, the lateral area of the pyramid is 472.5 in2. 2 2 126 cm ; 175 cm

54.

SOLUTION: The pyramid has a slant height of 15 inches and the Therefore, the surface area of the pyramid is about base is a hexagon with sides of 10.5 inches. 758.9 in2. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°. ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

Use a trigonometric ratio to find the measure of the apothem a. SOLUTION:

The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: Find the lateral area and surface area of the 205 in2 pyramid. Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

So, the lateral area of the pyramid is 472.5 in2.

The diameter of the circle is about 8.3 m. Therefore, the surface area of the pyramid is about ANSWER: 758.9 in2. 8.3 m

ANSWER: 57. Find the diameter of a circle with an area of 102 2 2 472.5 in ; 758.9 in square centimeters. SOLUTION: 55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

SOLUTION:

The area that needs to be coated is the sum of the The diameter of the circle is about 11.4 m. lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + ANSWER: 2 9)(2) + 13(9) = 205 in . 11.4 cm ANSWER: 58. The area of a circle is 191 square feet. Find the 2 205 in radius. SOLUTION: Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

ANSWER: 7.8 ft

59. Find the radius of a circle with an area of 271 square inches. The diameter of the circle is about 8.3 m. SOLUTION: ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

ANSWER: 9.3 in.

Find the area of each trapezoid, rhombus, or kite. The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm

58. The area of a circle is 191 square feet. Find the 60. radius. SOLUTION: SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 7.8 ft ANSWER: 2 59. Find the radius of a circle with an area of 271 square 120 in inches. SOLUTION:

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 9.3 in.

Find the area of each trapezoid, rhombus, or kite. ANSWER: 2 378 m

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

Find the volume of each prism.

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have 1. the same volume. So, the volume of a right prism and SOLUTION: an oblique one of the same height and cross sectional area are same. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 ANSWER: The volume is 108 cm . 3 26.95 m

ANSWER: 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters 108 cm3 SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

5. ANSWER: 3 SOLUTION: 396 in

3. the oblique rectangular prism shown at the right

SOLUTION: ANSWER: If two solids have the same height h and the same 206.4 ft3 cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

6. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross ANSWER: sectional area are same. 26.95 m3

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: 3 an oblique one of the same height and cross sectional 1357.2 m area are same. 7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION:

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the ANSWER: nearest tenth. 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches 5. SOLUTION: SOLUTION:

ANSWER: 3 410.1 in ANSWER: MULTIPLE CHOICE 3 9. A rectangular lap pool 206.4 ft measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400

6. C 30,000 SOLUTION: D 48,000 If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: ANSWER: D 3 1357.2 m CCSS SENSE-MAKING Find the volume of 7. a cylinder with a diameter of 16 centimeters and a each prism. height of 5.1 centimeters SOLUTION: 10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

ANSWER: 3 1025.4 cm

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches ANSWER: SOLUTION: 30 in3

11. ANSWER: 3 SOLUTION: 410.1 in The base is a triangle with a base length of 11 m and 9. MULTIPLE CHOICE A rectangular lap pool the corresponding height of 7 m. The height of the measures 80 feet long by 20 feet wide. If it needs to prism is 14 m. be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000 D 48,000 ANSWER: 3 539 m SOLUTION:

Each cubic foot holds 7.5 gallons of water. So, the 12. amount of water required to fill the pool is 6400(7.5) = 48,000. SOLUTION: The base is a right triangle with a leg length of 9 cm Therefore, the correct choice is D. and the hypotenuse of length 15 cm.

ANSWER: Use the Pythagorean Theorem to find the height of D the base.

CCSS SENSE-MAKING Find the volume of each prism.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. The height of the prism is 6 cm.

ANSWER: ANSWER: 3 3 30 in 324 cm

11. 13. SOLUTION: SOLUTION: The base is a triangle with a base length of 11 m and If two solids have the same height h and the same the corresponding height of 7 m. The height of the cross-sectional area B at every level, then they have prism is 14 m. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 ANSWER: B = 11.4 ft and h = 5.1 ft. Therefore, the volume is 3 539 m ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 12. centimeters and with a base area of 136 square centimeters SOLUTION: SOLUTION: The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and Use the Pythagorean Theorem to find the height of an oblique one of the same height and cross sectional the base. area are same.

ANSWER: 3 2040 cm

The height of the prism is 6 cm. 15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 324 cm3 area are same.

13. ANSWER: SOLUTION: 1534.25 in3 If two solids have the same height h and the same cross-sectional area B at every level, then they have CCSS SENSE-MAKING Find the volume of the same volume. So, the volume of a right prism and each cylinder. Round to the nearest tenth. an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the 16. area of a base and h is the height of the prism. SOLUTION: 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is r = 5 yd and h = 18 yd

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square ANSWER: 3 centimeters 1413.7 yd SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 17. area are same. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 2040 cm ANSWER: 15. a square prism with a base edge of 9.5 inches and a height of 17 inches 407.2 cm3 SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 18. area are same. SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

ANSWER: 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. SOLUTION: Now you can find the volume. r = 5 yd and h = 18 yd

ANSWER: ANSWER: 12-4 Volumes3 of Prisms and Cylinders 1413.7 yd 3 823.0 in

17. 19. SOLUTION: SOLUTION: r = 6 cm and h = 3.6 cm. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm.

ANSWER: 407.2 cm3

ANSWER: 18. 2686.1 mm3 SOLUTION: r = 5.5 in. 20. PLANTER A planter is in the shape of a

rectangular prism 18 inches long, inches deep, Use the Pythagorean Theorem to find the height of the cylinder. and 12 inches high. What is the volume of potting soil

in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER: 3 2740.5 in

21. SHIPPING A box 18 centimeters by 9 centimeters ANSWER: by 15 centimeters is being used to ship two 3 823.0 in cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown eSolutions Manual - Powered by Cognero at the right. What is the volume of the empty spacePage 5 in the box?

19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional SOLUTION: area are same. The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders. r = 7.5 mm and h = 15.2 mm.

ANSWER: ANSWER: 3 3 2686.1 mm 521.5 cm

20. PLANTER A planter is in the shape of a 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, rectangular prism 18 inches long, inches deep, and 10 cubic feet of sand. To transport the correct and 12 inches high. What is the volume of potting soil amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. in the planter if the planter is filled to inches What should the diameter of the cylinders be? below the top? SOLUTION: SOLUTION: 3 V = 10 ft and h = 2 ft Use the formula to find r. The planter is to be filled inches below the top, so

Therefore, the diameter of the cylinders should be about 2.52 ft. ANSWER: 3 ANSWER: 2740.5 in 2.52 ft 21. SHIPPING A box 18 centimeters by 9 centimeters Find the volume of the solid formed by each by 15 centimeters is being used to ship two net. cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box?

23. SOLUTION: The middle piece of the net is the front of the solid. SOLUTION: The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a The volume of the empty space is the difference of triangular prism. volumes of the rectangular prism and the cylinders.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm.

ANSWER: The volume V of a prism is V = Bh, where B is the 3 area of the base, h is the height of the prism. 521.5 cm 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? ANSWER: 3 SOLUTION: 3934.9 cm

3 V = 10 ft and h = 2 ft Use the formula to find r.

24.

Therefore, the diameter of the cylinders should be SOLUTION: about 2.52 ft. The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece ANSWER: were a rectangle, then the prism would be right. 2.52 ft However, since the middle piece is a parallelogram, it is oblique. Find the volume of the solid formed by each net. The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base. ANSWER: 3 48.9 m

25. FOOD A cylindrical can of baked potato chips has a The height of the prism is 20 cm. height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% The volume V of a prism is V = Bh, where B is the larger than the regular can. If both cans have the area of the base, h is the height of the prism. same radius, what is the height of the larger can?

ANSWER: SOLUTION: 3 The volume of the smaller can is 3934.9 cm The volume of the new can is 130% of the smaller can, with the same radius.

24.

SOLUTION: The circular bases at the top and bottom of the net The height of the new can will be 35.1 cm. indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. ANSWER: However, since the middle piece is a parallelogram, it 35.1 cm is oblique. 26. CHANGING DIMENSIONS A cylinder has a The radius is 1.8 m, the height is 4.8 m, and the slant radius of 5 centimeters and a height of 8 centimeters. height is 6 m. Describe how each change affects the volume of the cylinder. If two solids have the same height h and the same a. The height is tripled. cross-sectional area B at every level, then they have b. The radius is tripled. the same volume. So, the volume of a right prism and c. Both the radius and the height are tripled. an oblique one of the same height and cross sectional d. The dimensions are exchanged. area are same. SOLUTION:

a. When the height is tripled, h = 3h. ANSWER: 3 48.9 m

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius. So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER:

35.1 cm When the height and the radius are tripled, the

26. CHANGING DIMENSIONS A cylinder has a volume is multiplied by 27. radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the d. When the dimensions are exchanged, r = 8 and h cylinder. = 5 cm. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged.

SOLUTION: Compare to the original volume.

The volume is multiplied by . a. When the height is tripled, h = 3h. ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk When the height is tripled, the volume is multiplied by density of a potting soil to assess how well a specific 3. plant will grow in it. The density b. When the radius is tripled, r = 3r. of the soil sample is the ratio of its weight to its volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. So, when the radius is tripled, the volume is multiplied c. If a bag of this soil holds 2.5 cubic feet, what is its by 9. weight in pounds? SOLUTION: c. When the height and the radius are tripled, r = 3r and h = 3h.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm. The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus:

Compare to the original volume.

3 3 b. 0.0018 lb/in is close to 0.0019 lb/in so the plant The volume is multiplied by . should grow fairly well.

ANSWER: c. a. The volume is multiplied by 3. b. 2 The volume is multiplied by 3 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk ANSWER: density of a potting soil to assess how well a specific 3 plant will grow in it. The density a. 0.0019 lb / in of the soil sample is the ratio of its weight to its volume. b. The plant should grow well in this soil since the 3 bulk density of 0.0019 lb / in is close to the desired 3 a. If the weight of the container with the soil is 20 bulk density of 0.0018 lb / in . pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming c. 8.3 lb that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound Find the volume of each composite solid. Round per square inch is desirable for root growth? Explain. to the nearest tenth if necessary. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION:

28. SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm a. First calculate the volume of soil in the pot. Then and the height is 11 cm. The base of the other prism divide the weight of the soil by the volume. is 4 cm by 3 cm and the height is 5 cm.

The weight of the soil is the weight of the pot with ANSWER: soil minus the weight of the pot. 3 W = 20 – 5 = 15 lbs. 225 cm

The soil density is thus:

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. 29. c. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the 3 bulk density of 0.0019 lb / in is close to the desired 3 bulk density of 0.0018 lb / in . ANSWER: 3 c. 8.3 lb 120 m Find the volume of each composite solid. Round to the nearest tenth if necessary.

30. SOLUTION: 28. The solid is a combination of a rectangular prism and SOLUTION: two half cylinders.

The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: ANSWER: 260.5 in3 225 cm3 31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms. SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder. ANSWER: 3 120 m

Therefore, the volume of the rubberized material is about 304.1 cm3.

30. ANSWER: 3 SOLUTION: 304.1 cm The solid is a combination of a rectangular prism and Find each measure to the nearest tenth. two half cylinders. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the ANSWER: thickness of the base of the holder. The thickness of 5.7 cm the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER:

3 304.1 cm

Find each measure to the nearest tenth. ANSWER: 32. A cylindrical can has a volume of 363 cubic 3 centimeters. The diameter of the can is 9 678.6 in centimeters. What is the height? 34. A rectangular prism has a surface area of 432 SOLUTION: square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER: 5.7 cm Find the volume.

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

The volume will be the highest when the diameter is The radius is 6. Find the volume. 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER:

678.6 in3

34. A rectangular prism has a surface area of 432 ANSWER: square inches, a height of 6 inches, and a width of 12 3,190,680.0 cm3 inches. What is the volume? SOLUTION: 36. CCSS MODELING The base of a rectangular Use the surface area formula to find the length of the swimming pool is sloped so one end of the pool is 6 base of the prism. feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

SOLUTION: Find the volume. The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms.

ANSWER: 3 576 in

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest ANSWER: and smallest possible column? Round to the nearest 3 1575 ft tenth cubic centimeter. SOLUTION: 37. CHANGING DIMENSIONS A soy milk company The volume will be the highest when the diameter is is planning a promotion in which the volume of soy 95 cm and will be the lowest when it is 30 cm.That is milk in each container will be increased by 25%. The when the radii are 47.5 cm and 15 cm respectively. company wants the base of the container to stay the same. What will be the height of the new containers? Find the difference between the volumes.

SOLUTION: Find the volume of the original container.

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular The volume of the new container is 125% of the swimming pool is sloped so one end of the pool is 6 original container, with the same base dimensions. feet deep and the other end is 3 feet deep, as shown Use 1.25V and B to find h. in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

SOLUTION: The swimming pool is a combination of a rectangular ANSWER: prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height 38. DESIGN Sketch and label (in inches) three different of the trapezoidal prism is 15 ft. The total volume of designs for a dry ingredient measuring cup that holds the solid is the sum of the volumes of the two 1 cup. Be sure to include the dimensions in each prisms. drawing. (1 cup ≈ 14.4375 in3)

SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

ANSWER: 3 1575 ft

37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy The last equation gives us a relation between the milk in each container will be increased by 25%. The radius and height of the cylinder that must be fulfilled company wants the base of the container to stay the to get the desired volume. First, choose a suitable same. What will be the height of the new containers? radius, say 1.85 in, and solve for the height.

SOLUTION: If we choose a height of say 4 in., then we can solve Find the volume of the original container. for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

For any rectangular container, the volume equation is:

ANSWER: Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in. 38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius. ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 ANSWER: or 22 cm and a height of 10 cm. Sample answers: 40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism. The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option.

ANSWER: 3 SOLUTION: Because 2.96 yd of concrete are needed, the

The base of the prism can be divided into 5 second contractor is less expensive at $2181.50. congruent triangles of a base 8 cm and the 41. MULTIPLE REPRESENTATIONS In this corresponding height 5.5 cm. So, the pentagonal problem, you will investigate right and oblique prism is a combination of 5 triangular prisms of height cylinders 10 cm. Find the base area of each triangular prism. .

a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a Therefore, the volume of the pentagonal prism is diameter of 6 meters.

b. VERBAL A square prism has a height of 10 ANSWER: meters and a base edge of 6 meters. Is its volume 3 1100 cm ; Each triangular prism has a base area of greater than, less than, or equal to the volume of 2 the cylinder? Explain. or 22 cm and a height of 10 cm. c. ANALYTICAL Describe which change affects 40. PATIOS Mr. Thomas is planning to remove an old the volume of the cylinder more: multiplying the patio and install a new rectangular concrete patio 20 height by x or multiplying the radius by x. Explain. feet long, 12 feet wide, and 4 inches thick. One SOLUTION: contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio a. The oblique cylinder should look like the right and $700 for removal of the old patio. Which is the cylinder (same height and size), except that it is less expensive option? Explain. pushed a little to the side, like a slinky. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd

12 feet = 4 yd 4 in. = yd b. Find the volume of each.

Find the volume.

The total cost for the second contractor is about The volume of the square prism is greater. . c. Do each scenario.

Therefore, the second contractor is a less expensive option.

ANSWER: 3 Because 2.96 yd of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders .

a. GEOMETRIC Draw a right cylinder and an Assuming x > 1, multiplying the radius by x makes oblique cylinder with a height of 10 meters and a 2 diameter of 6 meters. the volume x times greater.

pushed a little to the side, like a slinky. b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater.

c. Multiplying the radius by x ; since the volume is b. Find the volume of each. π represented by r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

The volume of the square prism is greater. c. Do each scenario. SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater. ANSWER:

For example, if x = 0.5, then x2 = 0.25, which is less Francisco; Valerie incorrectly used as the than x. length of one side of the triangular base. Francisco used a different approach, but his solution is correct. ANSWER: a. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism

b. Greater than; a square with a side length of 6 m c. triangular prism with a right triangle as the base 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater.

c. Multiplying the radius by x ; since the volume is SOLUTION: represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the 3 radius by x makes the volume x 2 times greater, The volume of the can is 20π in . It takes three full assuming x > 1. cans to fill the container, so the volume of the container is 60π in 3. 42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular a. Choose some basic values for 2 of the sides, and prism with an apothem of 4 units and height of 5 then determine the third side. Base: 3 by 5. units. Is either of them correct? Explain your reasoning.

SOLUTION: 3 by 5 by 4π Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco b. Choose some basic values for 2 of the sides, and used a different approach, but his solution is correct. then determine the third side. Base: 5 by 5.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

ANSWER: 5 by 5 by Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco c. Choose some basic values for 2 of the sides, and used a different approach, but his solution is correct. then determine the third side. Base: Legs: 3 by 4. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism 3 by 4 by 10π

c. triangular prism with a right triangle as the base ANSWER: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by SOLUTION: c. base with legs measuring 3 in. and 4 in., height 3 10π in. The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the 44. WRITING IN MATH Write a helpful response to container is 60π in 3. the following question posted on an Internet gardening forum. a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your 3 by 5 by 4π garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. b. Choose some basic values for 2 of the sides, and ANSWER: then determine the third side. Base: 5 by 5. Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters.

SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest 5 by 5 by values to choose.

c. Choose some basic values for 2 of the sides, and Sample answer: then determine the third side. Base: Legs: 3 by 4.

ANSWER: Sample answer:

3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π 46. REASONING Determine whether the following statement is true or false. Explain. Two cylinders with the same height and the same b. 5 by 5 by lateral area must have the same volume.

c. base with legs measuring 3 in. and 4 in., height SOLUTION: 10π in. True; if two cylinders have the same height (h1 = h2) 44. WRITING IN MATH Write a helpful response to and the same lateral area (L1 = L2), the circular the following question posted on an Internet bases must have the same area. gardening forum.

volume of 50 cubic centimeters. SOLUTION: SOLUTION: Both formulas involve multiplying the area of the Choose 3 values that multiply to make 50. The base by the height. The base of a prism is a polygon, factors of 50 are 2, 5, 5, so these are the simplest so the expression representing the area varies, values to choose. depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2. Sample answer: ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr . ANSWER: Sample answer: 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm C 17 cm 46. REASONING Determine whether the following D 11.5 cm statement is true or false. Explain. SOLUTION: Two cylinders with the same height and the same

lateral area must have the same volume. SOLUTION:

True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area ANSWER: varies, depending on the type of polygon it is. The F 2 base of a cylinder is a circle, so its area is πr . 50. SHORT RESPONSE What is the ratio of the area 48. The volume of a triangular prism is 1380 cubic of the circle to the area of the square? centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm SOLUTION: C 17 cm The radius of the circle is 2x and the length of each D 11.5 cm side of the square is 4x. So, the ratio of the areas can SOLUTION: be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? ANSWER: A $4.25 B B $170 C $425 49. A cylindrical tank used for oil storage has a height D $4250 that is half the length of its radius. If the volume of E $42,500 the tank is 1,122,360 ft3, what is the tank’s radius? SOLUTION: F 89.4 ft G 178.8 ft Find the 0.5% of $85,000.

H 280.9 ft J 561.8 ft Therefore, the correct choice is C. SOLUTION: ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , ANSWER: where is the slant height and P is the perimeter of F the base.

50. SHORT RESPONSE What is the ratio of the area The slant height is the height of each of the of the circle to the area of the square? congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 Find the perimeter and area of the equilateral D $4250 triangle for the base. Use the Pythagorean Theorem E $42,500 to find the height h of the triangle. SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if

necessary.

52.

SOLUTION: The perimeter is P = 3 × 10 or 30 feet.

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the

Pythagorean Theorem to find the slant height. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

Find the perimeter and area of the equilateral Therefore, the surface area of the pyramid is about 2 triangle for the base. Use the Pythagorean Theorem 255.4 ft . to find the height h of the triangle. ANSWER: 212.1 ft2; 255.4 ft2

53. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

The perimeter is P = 3 × 10 or 30 feet.

So, the lateral area of the pyramid is 126 cm2.

So, the area of the base B is ft2. The surface area S of a regular pyramid is , where Find the lateral area L and surface area S of the L is the lateral area and B is the area of the base. regular pyramid.

Therefore, the surface area of the pyramid is 175 2 So, the lateral area of the pyramid is about 212.1 ft . cm2.

ANSWER: 2 2 126 cm ; 175 cm

54. Therefore, the surface area of the pyramid is about 2 255.4 ft . SOLUTION: The pyramid has a slant height of 15 inches and the ANSWER: base is a hexagon with sides of 10.5 inches. 212.1 ft2; 255.4 ft2 A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

53. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of Use a trigonometric ratio to find the measure of the the base. apothem a. Here, the base is a square of side 7 cm and the slant height is 9 cm.

Find the lateral area and surface area of the 2 So, the lateral area of the pyramid is 126 cm . pyramid.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 Therefore, the surface area of the pyramid is about 126 cm ; 175 cm 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2 54. 55. BAKING Many baking pans are given a special SOLUTION: nonstick coating. A rectangular cake pan is 9 inches The pyramid has a slant height of 15 inches and the by 13 inches by 2 inches deep. What is the area of base is a hexagon with sides of 10.5 inches. the inside of the pan that needs to be coated? A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 2 Use a trigonometric ratio to find the measure of the 205 in apothem a. Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

Find the lateral area and surface area of the pyramid.

So, the lateral area of the pyramid is 472.5 in2. The diameter of the circle is about 8.3 m. ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

SOLUTION: SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the ANSWER: diameter. 7.8 ft SOLUTION: 59. Find the radius of a circle with an area of 271 square inches. SOLUTION:

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m ANSWER: 57. Find the diameter of a circle with an area of 102 9.3 in. square centimeters. SOLUTION: Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the

lengths of its diagonals, d1 and d2. The diameter of the circle is about 11.4 m.

ANSWER: d1 = 12 in. and d2 = 7 + 13 = 20 in. 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 2 120 in

ANSWER: 61. 7.8 ft SOLUTION: The area A of a trapezoid is one half the product of 59. Find the radius of a circle with an area of 271 square the height h and the sum of the lengths of its bases, inches. b1 and b2. SOLUTION:

ANSWER: 2 378 m

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

Find the volume of each prism. 2. SOLUTION: The volume V of a prism is V = Bh, where B is the 1. area of a base and h is the height of the prism.

SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm . ANSWER: 3 396 in

ANSWER: 3. the oblique rectangular prism shown at the right 108 cm3

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have 2. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional SOLUTION: area are same. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER:

3 26.95 m

4. an oblique pentagonal prism with a base area of 42 ANSWER: square centimeters and a height of 5.2 centimeters 3 396 in SOLUTION: If two solids have the same height h and the same 3. the oblique rectangular prism shown at the right cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 5. 3 26.95 m SOLUTION:

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 206.4 ft3

6. ANSWER: SOLUTION: 3 218.4 cm If two solids have the same height h and the same cross-sectional area B at every level, then they have Find the volume of each cylinder. Round to the the same volume. So, the volume of a right cylinder nearest tenth. and an oblique one of the same height and cross sectional area are same.

5. SOLUTION:

ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters ANSWER: SOLUTION: 206.4 ft3

6. ANSWER: SOLUTION: 1025.4 cm3 If two solids have the same height h and the same cross-sectional area B at every level, then they have 8. a cylinder with a radius of 4.2 inches and a height of the same volume. So, the volume of a right cylinder 7.4 inches and an oblique one of the same height and cross SOLUTION: sectional area are same.

ANSWER: 3 ANSWER: 410.1 in 3 1357.2 m 9. MULTIPLE CHOICE A rectangular lap pool 7. a cylinder with a diameter of 16 centimeters and a measures 80 feet long by 20 feet wide. If it needs to height of 5.1 centimeters be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the SOLUTION: lap pool? A 4000 B 6400 C 30,000 D 48,000 SOLUTION: ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) SOLUTION: = 48,000.

Therefore, the correct choice is D.

ANSWER: D CCSS SENSE-MAKING Find the volume of ANSWER: each prism. 3 410.1 in

9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to 10. be filled to four feet deep and each cubic foot holds SOLUTION: 7.5 gallons, how many gallons will it take to fill the lap pool? The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. A 4000

B 6400 C 30,000 D 48,000

SOLUTION:

ANSWER: 30 in3

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. 11. ANSWER: SOLUTION: D The base is a triangle with a base length of 11 m and CCSS SENSE-MAKING Find the volume of the corresponding height of 7 m. The height of the each prism. prism is 14 m.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. ANSWER:

The height of the prism is 5 in. 3 539 m

ANSWER: 12. 3 30 in SOLUTION: The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm.

Use the Pythagorean Theorem to find the height of the base.

11. SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the prism is 14 m.

The height of the prism is 6 cm.

ANSWER: 3 539 m ANSWER: 324 cm3

12. SOLUTION: The base is a right triangle with a leg length of 9 cm

and the hypotenuse of length 15 cm. 13.

Use the Pythagorean Theorem to find the height of SOLUTION: the base. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is The height of the prism is 6 cm.

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 ANSWER: centimeters and with a base area of 136 square centimeters 324 cm3 SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 13. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 3 2040 cm The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 15. a square prism with a base edge of 9.5 inches and a height of 17 inches 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and 3 an oblique one of the same height and cross sectional 58.14 ft area are same. 14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and 3 an oblique one of the same height and cross sectional 1534.25 in area are same. CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. ANSWER: SOLUTION: 3 2040 cm r = 5 yd and h = 18 yd 15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 3 1413.7 yd area are same.

17.

SOLUTION: ANSWER: r = 6 cm and h = 3.6 cm. 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. ANSWER: 407.2 cm3 SOLUTION: r = 5 yd and h = 18 yd

18. SOLUTION: ANSWER: r = 5.5 in. 3 1413.7 yd Use the Pythagorean Theorem to find the height of the cylinder.

17. SOLUTION: r = 6 cm and h = 3.6 cm.

Now you can find the volume.

ANSWER: 407.2 cm3

ANSWER: 18. 3 823.0 in SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

r = 7.5 mm and h = 15.2 mm.

Now you can find the volume.

ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a ANSWER: 3 rectangular prism 18 inches long, inches deep, 823.0 in and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? 19. SOLUTION: SOLUTION: The planter is to be filled inches below the top, so If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm.

ANSWER: 3 2740.5 in

21. SHIPPING A box 18 centimeters by 9 centimeters ANSWER: by 15 centimeters is being used to ship two 3 2686.1 mm cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown 20. PLANTER A planter is in the shape of a at the right. What is the volume of the empty space in the box? rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top?

SOLUTION: SOLUTION: The planter is to be filled inches below the top, so The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

ANSWER: ANSWER: 12-4 Volumes of Prisms and Cylinders 3 2740.5 in 521.5 cm3

21. SHIPPING A box 18 centimeters by 9 centimeters 22. SANDCASTLES In a sandcastle competition, by 15 centimeters is being used to ship two contestants are allowed to use only water, shovels, cylindrical candles. Each candle has a diameter of 9 and 10 cubic feet of sand. To transport the correct centimeters and a height of 15 centimeters, as shown amount of sand, they want to create cylinders that at the right. What is the volume of the empty space are 2 feet tall to hold enough sand for one contestant. in the box? What should the diameter of the cylinders be? SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER:

2.52 ft

ANSWER: Find the volume of the solid formed by each net. 521.5 cm3

22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? 23. SOLUTION: SOLUTION: The middle piece of the net is the front of the solid. 3 The top and bottom pieces are the bases and the V = 10 ft and h = 2 ft Use the formula to find r. pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

Therefore, the diameter of the cylinders should be The height of the prism is 20 cm. about 2.52 ft. The volume V of a prism is V = Bh, where B is the ANSWER: area of the base, h is the height of the prism. 2.52 ft eSolutionsFindManual the volume- Powered ofby theCognero solid formed by each Page 6 net.

ANSWER: 3934.9 cm3

23. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 24. cm. Use the Pythagorean Theorem to find the height of the base. SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. The height of the prism is 20 cm. However, since the middle piece is a parallelogram, it is oblique. The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism. The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

3934.9 cm3

ANSWER: 3 48.9 m 24. 25. FOOD A cylindrical can of baked potato chips has a SOLUTION: height of 27 centimeters and a radius of 4 The circular bases at the top and bottom of the net centimeters. A new can is advertised as being 30% indicate that this is a cylinder. If the middle piece larger than the regular can. If both cans have the were a rectangle, then the prism would be right. same radius, what is the height of the larger can? However, since the middle piece is a parallelogram, it is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant

height is 6 m.

If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have The volume of the smaller can is the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional The volume of the new can is 130% of the smaller area are same. can, with the same radius.

ANSWER:

3 48.9 m The height of the new can will be 35.1 cm.

25. FOOD A cylindrical can of baked potato chips has a ANSWER: height of 27 centimeters and a radius of 4 35.1 cm centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the 26. CHANGING DIMENSIONS A cylinder has a same radius, what is the height of the larger can? radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. SOLUTION: d. The dimensions are exchanged. The volume of the smaller can is SOLUTION:

The volume of the new can is 130% of the smaller can, with the same radius.

a. When the height is tripled, h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: When the height is tripled, the volume is multiplied by 35.1 cm 3. b. 26. CHANGING DIMENSIONS A cylinder has a When the radius is tripled, r = 3r. radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged.

SOLUTION: So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

a. When the height is tripled, h = 3h.

When the height and the radius are tripled, the volume is multiplied by 27.

When the height is tripled, the volume is multiplied by d. When the dimensions are exchanged, r = 8 and h 3. = 5 cm. b. When the radius is tripled, r = 3r.

Compare to the original volume.

So, when the radius is tripled, the volume is multiplied by 9. The volume is multiplied by .

c. When the height and the radius are tripled, r = 3r and h = 3h. ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific When the height and the radius are tripled, the plant will grow in it. The density volume is multiplied by 27. of the soil sample is the ratio of its weight to its volume. d. When the dimensions are exchanged, r = 8 and h = 5 cm. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its

weight in pounds? Compare to the original volume. SOLUTION:

The volume is multiplied by .

ANSWER:

a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. a. First calculate the volume of soil in the pot. Then 3 divide the weight of the soil by the volume. c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its The weight of the soil is the weight of the pot with volume. soil minus the weight of the pot. W = 20 – 5 = 15 lbs. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 The soil density is thus: pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain.

c. If a bag of this soil holds 2.5 cubic feet, what is its 3 3 weight in pounds? b. 0.0018 lb/in is close to 0.0019 lb/in so the plant should grow fairly well. SOLUTION: c.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the 3 bulk density of 0.0019 lb / in is close to the desired 3 bulk density of 0.0018 lb / in . The weight of the soil is the weight of the pot with soil minus the weight of the pot. c. 8.3 lb W = 20 – 5 = 15 lbs. Find the volume of each composite solid. Round The soil density is thus: to the nearest tenth if necessary.

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. 28. SOLUTION: c. The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: 3 a. 0.0019 lb / in

ANSWER: b. The plant should grow well in this soil since the 3 bulk density of 0.0019 lb / in3 is close to the desired 225 cm 3 bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

29. SOLUTION: 28. The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid SOLUTION: is the sum of the volumes of the two rectangular The solid is a combination of two rectangular prisms. prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: ANSWER: 3 225 cm3 120 m

30. SOLUTION: The solid is a combination of a rectangular prism and 29. two half cylinders. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 260.5 in

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 ANSWER: centimeters tall, including 1 centimeter for the 3 120 m thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

30. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm.

The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the ANSWER: radius of the base and h is the height of the cylinder. 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5

centimeters tall, including 1 centimeter for the

thickness of the base of the holder. The thickness of Therefore, the volume of the rubberized material is 3 the rim of the holder is 1 centimeter. What is the about 304.1 cm . volume of the rubberized material that makes up the holder? ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION: SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder. ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches Therefore, the volume of the rubberized material is and a height of 6 inches. What is the volume? 3 about 304.1 cm . SOLUTION: ANSWER: Use the surface area formula to solve for r.

3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

The radius is 6. Find the volume.

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume?

SOLUTION: Use the surface area formula to solve for r. ANSWER: 678.6 in3

The radius is 6. Find the volume. Find the volume.

ANSWER: ANSWER: 3 576 in 678.6 in3 35. ARCHITECTURE A cylindrical stainless steel 34. A rectangular prism has a surface area of 432 column is used to hide a ventilation system in a new square inches, a height of 6 inches, and a width of 12 building. According to the specifications, the diameter inches. What is the volume? of the column can be between 30 centimeters and 95 SOLUTION: centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest Use the surface area formula to find the length of the and smallest possible column? Round to the nearest base of the prism. tenth cubic centimeter.

SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

Find the volume.

ANSWER: ANSWER: 3 576 in 3,190,680.0 cm3

35. ARCHITECTURE A cylindrical stainless steel 36. CCSS MODELING The base of a rectangular column is used to hide a ventilation system in a new swimming pool is sloped so one end of the pool is 6 building. According to the specifications, the diameter feet deep and the other end is 3 feet deep, as shown of the column can be between 30 centimeters and 95 in the figure. If the width is 15 feet, find the volume centimeters. The height is to be 500 centimeters. of water it takes to fill the pool. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is SOLUTION: when the radii are 47.5 cm and 15 cm respectively. The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the Find the difference between the volumes. rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms.

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 ANSWER: 3 feet deep and the other end is 3 feet deep, as shown 1575 ft in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool. 37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the same. What will be the height of the new containers?

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

ANSWER: 3 1575 ft

37. CHANGING DIMENSIONS A soy milk company

is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the ANSWER: same. What will be the height of the new containers?

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3)

SOLUTION: SOLUTION: Find the volume of the original container. Sample answers:

For any cylindrical container, we have the following equation for volume:

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each If we choose a height of say 4 in., then we can solve for the radius. drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

For any rectangular container, the volume equation is: The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height. Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

If we choose a height of say 4 in., then we can solve for the radius.

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

ANSWER: Sample answers:

39. Find the volume of the regular pentagonal prism by ANSWER: dividing it into five equal triangular prisms. Describe Sample answers: the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old 39. Find the volume of the regular pentagonal prism by patio and install a new rectangular concrete patio 20 dividing it into five equal triangular prisms. Describe feet long, 12 feet wide, and 4 inches thick. One the base area and height of each triangular prism. contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

SOLUTION: 20 feet = yd The base of the prism can be divided into 5 12 feet = 4 yd congruent triangles of a base 8 cm and the 4 in. = yd corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism. Find the volume.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

The total cost for the second contractor is about 40. PATIOS Mr. Thomas is planning to remove an old . patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One Therefore, the second contractor is a less expensive contractor bid $2225 for the project. A second option. contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the ANSWER: less expensive option? Explain. Because 2.96 yd3 of concrete are needed, the SOLUTION: second contractor is less expensive at $2181.50. Convert all of the dimensions to yards. 41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique 20 feet = yd cylinders 12 feet = 4 yd . 4 in. = yd a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a Find the volume. diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: The total cost for the second contractor is about a. The oblique cylinder should look like the right . cylinder (same height and size), except that it is pushed a little to the side, like a slinky. Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this b. Find the volume of each. problem, you will investigate right and oblique cylinders . a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects The volume of the square prism is greater. the volume of the cylinder more: multiplying the c. Do each scenario. height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Find the volume of each. Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less than x.

ANSWER:

The volume of the square prism is greater. c. Do each scenario.

Assuming x > 1, multiplying the radius by x makes assuming x > 1. 2 the volume x times greater. 42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular For example, if x = 0.5, then x2 = 0.25, which is less prism with an apothem of 4 units and height of 5 than x. units. Is either of them correct? Explain your reasoning. ANSWER: a.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

b. Greater than; a square with a side length of 6 m 2 Francisco used the standard formula for the volume has an area of 36 m . A circle with a diameter of 6 of a solid, V = Bh. The area of the base, B, is one- m has an area of 9π or 28.3 m2. Since the heights half the apothem multiplied by the perimeter of the are the same, the volume of the square prism is base. greater.

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the ANSWER: radius by x makes the volume x 2 times greater, assuming x > 1. Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco

42. CCSS CRITIQUE Franciso and Valerie each used a different approach, but his solution is correct. calculated the volume of an equilateral triangular 43. CHALLENGE A cylindrical can is used to fill a prism with an apothem of 4 units and height of 5 container with liquid. It takes three full cans to fill the units. Is either of them correct? Explain your container. Describe possible dimensions of the reasoning. container if it is each of the following shapes.

a. rectangular prism

b. square prism SOLUTION: Francisco; Valerie incorrectly used as the c. triangular prism with a right triangle as the base length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base. SOLUTION:

3 The volume of the can is 20π in . It takes three full

cans to fill the container, so the volume of the

container is 60π in 3. ANSWER: Francisco; Valerie incorrectly used as the a. Choose some basic values for 2 of the sides, and

length of one side of the triangular base. Francisco then determine the third side. Base: 3 by 5. used a different approach, but his solution is correct.

a. rectangular prism 3 by 5 by 4π

b. b. square prism Choose some basic values for 2 of the sides, and

then determine the third side. Base: 5 by 5. c. triangular prism with a right triangle as the base

SOLUTION:

3 The volume of the can is 20π in . It takes three full 5 by 5 by cans to fill the container, so the volume of the container is 60π in 3. c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5.

3 by 4 by 10π 3 by 5 by 4π ANSWER:

Sample answers: b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. a. 3 by 5 by 4π

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum. 5 by 5 by I am new to gardening. The nursery will deliver a c. Choose some basic values for 2 of the sides, and truckload of soil, which they say is 4 yards. I then determine the third side. Base: Legs: 3 by 4. know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

3 by 4 by 10π ANSWER: Sample answer: The nursery means a cubic yard, ANSWER: which is 33 or 27 cubic feet. Find the volume of your Sample answers: garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. a. 3 by 5 by 4π 45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. b. 5 by 5 by SOLUTION: c. base with legs measuring 3 in. and 4 in., height Choose 3 values that multiply to make 50. The 10π in. factors of 50 are 2, 5, 5, so these are the simplest values to choose. 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet Sample answer: gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? ANSWER: SOLUTION: Sample answer: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard, 3 46. REASONING Determine whether the following which is 3 or 27 cubic feet. Find the volume of your statement is true or false. Explain. garden in cubic feet and divide by 27 to determine Two cylinders with the same height and the same the number of cubic yards of soil needed. lateral area must have the same volume. 45. OPEN ENDED Draw and label a prism that has a SOLUTION: volume of 50 cubic centimeters. True; if two cylinders have the same height (h1 = h2) SOLUTION: and the same lateral area (L1 = L2), the circular Choose 3 values that multiply to make 50. The bases must have the same area. factors of 50 are 2, 5, 5, so these are the simplest values to choose.

Sample answer:

The radii must also be equal. ANSWER: ANSWER: Sample answer: True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 same area. Therefore, πr h is the same for each cylinder.

47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder REASONING 46. Determine whether the following similar? How are they different? statement is true or false. Explain. Two cylinders with the same height and the same SOLUTION: lateral area must have the same volume. Both formulas involve multiplying the area of the SOLUTION: base by the height. The base of a prism is a polygon, so the expression representing the area varies, True; if two cylinders have the same height (h1 = h2) depending on the type of polygon it is. The base of a 2 and the same lateral area (L1 = L2), the circular cylinder is a circle, so its area is πr . bases must have the same area. ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr .

48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs The radii must also be equal. measuring 8 centimeters and 15 centimeters. What is the height of the prism? ANSWER: A 34.5 cm True; if two cylinders have the same height and the B 23 cm same lateral area, the circular bases must have the C 17 cm 2 same area. Therefore, πr h is the same for each D 11.5 cm cylinder. SOLUTION: 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a 2 cylinder is a circle, so its area is πr .

ANSWER: ANSWER: Sample answer: Both formulas involve multiplying the B area of the base by the height. The base of a prism is a polygon, so the expression representing the area 49. A cylindrical tank used for oil storage has a height varies, depending on the type of polygon it is. The that is half the length of its radius. If the volume of 2 base of a cylinder is a circle, so its area is πr . the tank is 1,122,360 ft3, what is the tank’s radius? F 89.4 ft 48. The volume of a triangular prism is 1380 cubic G 178.8 ft centimeters. Its base is a right triangle with legs H measuring 8 centimeters and 15 centimeters. What is 280.9 ft the height of the prism? J 561.8 ft A 34.5 cm SOLUTION: B 23 cm C 17 cm D 11.5 cm SOLUTION:

ANSWER: ANSWER: F B 50. SHORT RESPONSE What is the ratio of the area 49. A cylindrical tank used for oil storage has a height of the circle to the area of the square? that is half the length of its radius. If the volume of the tank is 1,122,360 ft3, what is the tank’s radius? F 89.4 ft G 178.8 ft H 280.9 ft SOLUTION: J 561.8 ft The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can SOLUTION: be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 ANSWER: D $4250 F E $42,500 50. SHORT RESPONSE What is the ratio of the area SOLUTION: of the circle to the area of the square? Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: SOLUTION: C The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can Find the lateral area and surface area of each be written as shown. regular pyramid. Round to the nearest tenth if necessary.

ANSWER:

52.

51. SAT/ACT A county proposes to enact a new 0.5% SOLUTION: property tax. What would be the additional tax The lateral area L of a regular pyramid is , amount for a landowner whose property has a taxable value of $85,000? where is the slant height and P is the perimeter of A $4.25 the base. B $170 C $425 The slant height is the height of each of the D $4250 congruent lateral triangular faces. Use the E $42,500 Pythagorean Theorem to find the slant height. SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , Find the perimeter and area of the equilateral where is the slant height and P is the perimeter of triangle for the base. Use the Pythagorean Theorem the base. to find the height h of the triangle.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

The perimeter is P = 3 × 10 or 30 feet.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

2 So, the lateral area of the pyramid is about 212.1 ft .

The perimeter is P = 3 × 10 or 30 feet.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 212.1 ft2; 255.4 ft2

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

53. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of So, the lateral area of the pyramid is about 212.1 ft2. the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the lateral area of the pyramid is 126 cm2. Therefore, the surface area of the pyramid is about 2 255.4 ft . The surface area S of a regular ANSWER: pyramid is , where 2 2 212.1 ft ; 255.4 ft L is the lateral area and B is the area of the base.

53. Therefore, the surface area of the pyramid is 175 SOLUTION: cm2. The lateral area L of a regular pyramid is , ANSWER: 2 2 where is the slant height and P is the perimeter of 126 cm ; 175 cm the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

54. SOLUTION: The pyramid has a slant height of 15 inches and the So, the lateral area of the pyramid is 126 cm2. base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°. The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

Therefore, the surface area of the pyramid is 175 Use a trigonometric ratio to find the measure of the cm2. apothem a.

ANSWER: 2 2 126 cm ; 175 cm

Find the lateral area and surface area of the 54. pyramid.

SOLUTION: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°. So, the lateral area of the pyramid is 472.5 in2.

Use a trigonometric ratio to find the measure of the Therefore, the surface area of the pyramid is about apothem a. 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches Find the lateral area and surface area of the by 13 inches by 2 inches deep. What is the area of pyramid. the inside of the pan that needs to be coated?

SOLUTION: So, the lateral area of the pyramid is 472.5 in2. The area that needs to be coated is the sum of the

lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the indicated measure. Round to the nearest tenth. Therefore, the surface area of the pyramid is about 56. The area of a circle is 54 square meters. Find the diameter. 758.9 in2. SOLUTION:

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

The diameter of the circle is about 8.3 m.

ANSWER: SOLUTION: 8.3 m The area that needs to be coated is the sum of the lateral area and one base area. 57. Find the diameter of a circle with an area of 102 Therefore, the area that needs to be coated is 2(13 + square centimeters. 2 9)(2) + 13(9) = 205 in . SOLUTION: ANSWER: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters.

SOLUTION:

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches. SOLUTION:

The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius.

SOLUTION:

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

ANSWER: 60. 7.8 ft SOLUTION: 59. Find the radius of a circle with an area of 271 square The area A of a kite is one half the product of the inches. lengths of its diagonals, d1 and d2. SOLUTION: d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite. 61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, 60. b1 and b2. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 378 m

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

Find the volume of each prism.

SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

3 The volume is 108 cm . SOLUTION: ANSWER: If two solids have the same height h and the same cross-sectional area B at every level, then they have 3 108 cm the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

SOLUTION: The volume V of a prism is V = Bh, where B is the ANSWER: area of a base and h is the height of the prism. 26.95 m3

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

ANSWER: 3 218.4 cm

SOLUTION: Find the volume of each cylinder. Round to the nearest tenth. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

5. SOLUTION:

ANSWER: 3 26.95 m ANSWER: 4. an oblique pentagonal prism with a base area of 42 206.4 ft3 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 6. an oblique one of the same height and cross sectional area are same. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 3 1357.2 m

5. 7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION: SOLUTION:

ANSWER: ANSWER: 3 206.4 ft3 1025.4 cm 8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION:

A 4000 ANSWER: B 6400 3 C 1357.2 m 30,000 D 48,000 7. a cylinder with a diameter of 16 centimeters and a SOLUTION: height of 5.1 centimeters SOLUTION:

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

ANSWER: Therefore, the correct choice is D. 1025.4 cm3 ANSWER: D 8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches CCSS SENSE-MAKING Find the volume of SOLUTION: each prism.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. ANSWER: The height of the prism is 5 in. 3 410.1 in

11. SOLUTION: The base is a triangle with a base length of 11 m and Each cubic foot holds 7.5 gallons of water. So, the the corresponding height of 7 m. The height of the amount of water required to fill the pool is 6400(7.5) prism is 14 m. = 48,000.

Therefore, the correct choice is D.

ANSWER: D CCSS SENSE-MAKING Find the volume of ANSWER: 3 each prism. 539 m

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. 12. The height of the prism is 5 in. SOLUTION:

The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm.

Use the Pythagorean Theorem to find the height of the base.

ANSWER: 30 in3

11.

SOLUTION: The height of the prism is 6 cm.

The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the prism is 14 m.

ANSWER: 324 cm3

ANSWER: 3 539 m

13. SOLUTION: If two solids have the same height h and the same 12. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and SOLUTION: an oblique one of the same height and cross sectional The base is a right triangle with a leg length of 9 cm area are same. and the hypotenuse of length 15 cm. The volume V of a prism is V = Bh, where B is the Use the Pythagorean Theorem to find the height of area of a base and h is the height of the prism. the base. 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters The height of the prism is 6 cm. SOLUTION:

ANSWER: 3 2040 cm 13. SOLUTION: 15. a square prism with a base edge of 9.5 inches and a height of 17 inches If two solids have the same height h and the same cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right prism and If two solids have the same height h and the same an oblique one of the same height and cross sectional cross-sectional area B at every level, then they have area are same. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional The volume V of a prism is V = Bh, where B is the area are same. area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

ANSWER: 58.14 ft3 ANSWER: 1534.25 in3 14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square CCSS SENSE-MAKING Find the volume of centimeters each cylinder. Round to the nearest tenth. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 16. an oblique one of the same height and cross sectional SOLUTION: area are same. r = 5 yd and h = 18 yd

ANSWER: 3 ANSWER: 2040 cm 3 1413.7 yd 15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have 17. the same volume. So, the volume of a right prism and SOLUTION: an oblique one of the same height and cross sectional r = 6 cm and h = 3.6 cm. area are same.

ANSWER: ANSWER: 3 1534.25 in 407.2 cm3 CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

18. 16. SOLUTION: SOLUTION: r = 5.5 in. r = 5 yd and h = 18 yd Use the Pythagorean Theorem to find the height of the cylinder.

ANSWER: 3 1413.7 yd

17.

SOLUTION: Now you can find the volume.

r = 6 cm and h = 3.6 cm.

ANSWER: 3 ANSWER: 823.0 in 407.2 cm3

19.

18. SOLUTION: If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have r = 5.5 in. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional Use the Pythagorean Theorem to find the height of area are same. the cylinder. r = 7.5 mm and h = 15.2 mm.

ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a Now you can find the volume. rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top?

SOLUTION: ANSWER: 3 The planter is to be filled inches below the top, so 823.0 in

19. SOLUTION:

If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 3 area are same. 2740.5 in

21. SHIPPING A box 18 centimeters by 9 centimeters r = 7.5 mm and h = 15.2 mm. by 15 centimeters is being used to ship two

cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box?

ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a SOLUTION: rectangular prism 18 inches long, inches deep, The volume of the empty space is the difference of and 12 inches high. What is the volume of potting soil volumes of the rectangular prism and the cylinders.

in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

ANSWER: 521.5 cm3

21. SHIPPING A box 18 centimeters by 9 centimeters 3 by 15 centimeters is being used to ship two V = 10 ft and h = 2 ft Use the formula to find r. cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box?

Therefore, the diameter of the cylinders should be about 2.52 ft. SOLUTION: ANSWER: The volume of the empty space is the difference of 2.52 ft volumes of the rectangular prism and the cylinders. Find the volume of the solid formed by each net.

ANSWER: 23. 3 521.5 cm SOLUTION: The middle piece of the net is the front of the solid. 22. SANDCASTLES In a sandcastle competition, The top and bottom pieces are the bases and the contestants are allowed to use only water, shovels, pieces on the ends are the side faces. This is a and 10 cubic feet of sand. To transport the correct triangular prism. amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. One leg of the base 14 cm and the hypotenuse 31.4

What should the diameter of the cylinders be? cm. Use the Pythagorean Theorem to find the height SOLUTION: of the base.

3 V = 10 ft and h = 2 ft Use the formula to find r. The height of the prism is 20 cm.

The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism.

Therefore, the diameter of the cylinders should be about 2.52 ft. ANSWER: ANSWER: 12-4 Volumes of Prisms and Cylinders 3934.9 cm3 2.52 ft Find the volume of the solid formed by each net.

23. 24. SOLUTION: SOLUTION: The middle piece of the net is the front of the solid. The circular bases at the top and bottom of the net The top and bottom pieces are the bases and the indicate that this is a cylinder. If the middle piece pieces on the ends are the side faces. This is a were a rectangle, then the prism would be right. triangular prism. However, since the middle piece is a parallelogram, it is oblique. One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height The radius is 1.8 m, the height is 4.8 m, and the slant of the base. height is 6 m.

If two solids have the same height h and the same cross-sectional area B at every level, then they have The height of the prism is 20 cm. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional The volume V of a prism is V = Bh, where B is the area are same. area of the base, h is the height of the prism.

ANSWER: 3934.9 cm3 ANSWER: 3 48.9 m

24.

SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece SOLUTION: were a rectangle, then the prism would be right. The volume of the smaller can is However, since the middle piece is a parallelogram, it is oblique. The volume of the new can is 130% of the smaller eSolutions Manual - Powered by Cognero can, with the same radius. Page 7 The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. ANSWER: Describe how each change affects the volume of the 3 48.9 m cylinder. a. The height is tripled. 25. FOOD A cylindrical can of baked potato chips has a b. The radius is tripled. height of 27 centimeters and a radius of 4 c. Both the radius and the height are tripled. centimeters. A new can is advertised as being 30% d. The dimensions are exchanged. larger than the regular can. If both cans have the same radius, what is the height of the larger can? SOLUTION:

SOLUTION: a. The volume of the smaller can is When the height is tripled, h = 3h.

The volume of the new can is 130% of the smaller can, with the same radius.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. So, when the radius is tripled, the volume is multiplied Describe how each change affects the volume of the by 9. cylinder. a. The height is tripled. c. When the height and the radius are tripled, r = 3r b. The radius is tripled. and h = 3h. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

When the height and the radius are tripled, the a. volume is multiplied by 27. When the height is tripled, h = 3h. d. When the dimensions are exchanged, r = 8 and h = 5 cm.

When the height is tripled, the volume is multiplied by 3. Compare to the original volume.

b. When the radius is tripled, r = 3r.

The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27. So, when the radius is tripled, the volume is multiplied

by 9. d. The volume is multiplied by

c. When the height and the radius are tripled, r = 3r 27. SOIL A soil scientist wants to determine the bulk and h = 3h. density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a When the height and the radius are tripled, the plant grow in this soil if a bulk density of 0.018 pound volume is multiplied by 27. per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its d. When the dimensions are exchanged, r = 8 and h weight in pounds? = 5 cm. SOLUTION:

Compare to the original volume. a. First calculate the volume of soil in the pot. Then

divide the weight of the soil by the volume.

The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. 2 b. The volume is multiplied by 3 or 9. The weight of the soil is the weight of the pot with 3 c. The volume is multiplied by 3 or 27. soil minus the weight of the pot. W = 20 – 5 = 15 lbs. d. The volume is multiplied by The soil density is thus: 27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density

of the soil sample is the ratio of its weight to its volume. b 3 3 . 0.0018 lb/in is close to 0.0019 lb/in so the plant should grow fairly well. a. If the weight of the container with the soil is 20

pounds and the weight of the container alone is 5 c. pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION: ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 a. First calculate the volume of soil in the pot. Then bulk density of 0.0018 lb / in . divide the weight of the soil by the volume. c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs. 28.

The soil density is thus: SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

ANSWER: 225 cm3

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in . 29.

c. 8.3 lb SOLUTION: The solid is a combination of a rectangular prism and Find the volume of each composite solid. Round a right triangular prism. The total volume of the solid to the nearest tenth if necessary. is the sum of the volumes of the two rectangular prisms.

28. SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm. ANSWER: 3 120 m

ANSWER: 30. 225 cm3 SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

29. SOLUTION: The solid is a combination of a rectangular prism and ANSWER: a right triangular prism. The total volume of the solid 3 is the sum of the volumes of the two rectangular 260.5 in prisms. 31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

ANSWER: 3 120 m

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and 30. the space used for the can. The container has a SOLUTION: radius of and a height of 11.5 cm. The solid is a combination of a rectangular prism and two half cylinders. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is about 304.1 cm3. ANSWER: 3 ANSWER: 260.5 in 3 304.1 cm 31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 Find each measure to the nearest tenth. centimeters tall, including 1 centimeter for the 32. A cylindrical can has a volume of 363 cubic thickness of the base of the holder. The thickness of centimeters. The diameter of the can is 9 the rim of the holder is 1 centimeter. What is the centimeters. What is the height? volume of the rubberized material that makes up the SOLUTION: holder?

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a ANSWER: 5.7 cm radius of and a height of 11.5 cm. The empty space used to keep the can has a radius 33. A cylinder has a surface area of 144π square inches of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The and a height of 6 inches. What is the volume? 2 volume V of a cylinder is V = πr h, where r is the SOLUTION: radius of the base and h is the height of the cylinder. Use the surface area formula to solve for r.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height?

SOLUTION: The radius is 6. Find the volume.

ANSWER: ANSWER: 678.6 in3 5.7 cm 34. A rectangular prism has a surface area of 432 33. A cylinder has a surface area of 144π square inches square inches, a height of 6 inches, and a width of 12

and a height of 6 inches. What is the volume? inches. What is the volume? SOLUTION: SOLUTION: Use the surface area formula to solve for r. Use the surface area formula to find the length of the

base of the prism.

Find the volume.

The radius is 6. Find the volume.

ANSWER: 3 576 in

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest ANSWER: and smallest possible column? Round to the nearest 678.6 in3 tenth cubic centimeter. SOLUTION: 34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 The volume will be the highest when the diameter is inches. What is the volume? 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively. SOLUTION: Use the surface area formula to find the length of the Find the difference between the volumes. base of the prism.

Find the volume. ANSWER: 3,190,680.0 cm3

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter SOLUTION: of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. The swimming pool is a combination of a rectangular What is the difference in volume between the largest prism and a trapezoidal prism. The base of the and smallest possible column? Round to the nearest rectangular prism is 6 ft by 10 ft and the height is 15 tenth cubic centimeter. ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height SOLUTION: of the trapezoidal prism is 15 ft. The total volume of The volume will be the highest when the diameter is the solid is the sum of the volumes of the two 95 cm and will be the lowest when it is 30 cm.That is prisms. when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 3 1575 ft

37. CHANGING DIMENSIONS A soy milk company ANSWER: is planning a promotion in which the volume of soy 3,190,680.0 cm3 milk in each container will be increased by 25%. The company wants the base of the container to stay the 36. CCSS MODELING The base of a rectangular same. What will be the height of the new containers? swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

SOLUTION:

Find the volume of the original container. SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height The volume of the new container is 125% of the of the trapezoidal prism is 15 ft. The total volume of original container, with the same base dimensions. the solid is the sum of the volumes of the two Use 1.25V and B to find h. prisms.

ANSWER: ANSWER: 3 1575 ft 38. DESIGN Sketch and label (in inches) three different 37. CHANGING DIMENSIONS A soy milk company designs for a dry ingredient measuring cup that holds is planning a promotion in which the volume of soy 1 cup. Be sure to include the dimensions in each milk in each container will be increased by 25%. The 3 company wants the base of the container to stay the drawing. (1 cup ≈ 14.4375 in ) same. What will be the height of the new containers? SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

SOLUTION: Find the volume of the original container.

The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable The volume of the new container is 125% of the radius, say 1.85 in, and solve for the height. original container, with the same base dimensions. Use 1.25V and B to find h.

If we choose a height of say 4 in., then we can solve for the radius. ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following For any rectangular container, the volume equation is: equation for volume:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

If we choose a height of say 4 in., then we can solve for the radius.

For any rectangular container, the volume equation is: ANSWER:

Sample answers: Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

ANSWER: Sample answers: SOLUTION: The base of the prism can be divided into 5 congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

Find the volume.

Therefore, the volume of the pentagonal prism is The total cost for the second contractor is about .

ANSWER: Therefore, the second contractor is a less expensive 3 option. 1100 cm ; Each triangular prism has a base area of 2 ANSWER: or 22 cm and a height of 10 cm. Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50. 40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 41. MULTIPLE REPRESENTATIONS In this feet long, 12 feet wide, and 4 inches thick. One problem, you will investigate right and oblique contractor bid $2225 for the project. A second cylinders contractor bid $500 per cubic yard for the new patio . and $700 for removal of the old patio. Which is the a. GEOMETRIC Draw a right cylinder and an

less expensive option? Explain. oblique cylinder with a height of 10 meters and a SOLUTION: diameter of 6 meters. Convert all of the dimensions to yards. b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume 20 feet = yd greater than, less than, or equal to the volume of 12 feet = 4 yd the cylinder? Explain. 4 in. = yd c. ANALYTICAL Describe which change affects Find the volume. the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

The total cost for the second contractor is about . b. Find the volume of each. Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders . a. GEOMETRIC Draw a right cylinder and an

oblique cylinder with a height of 10 meters and a The volume of the square prism is greater. diameter of 6 meters. c. Do each scenario.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky. Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less than x.

ANSWER: a. b. Find the volume of each.

The volume of the square prism is greater. greater.

c. Do each scenario. c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less SOLUTION: than x. Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco ANSWER: used a different approach, but his solution is correct. a. Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 ANSWER: m has an area of 9π or 28.3 m2. Since the heights Francisco; Valerie incorrectly used as the are the same, the volume of the square prism is length of one side of the triangular base. Francisco greater. used a different approach, but his solution is correct.

c. Multiplying the radius by x ; since the volume is 43. CHALLENGE A cylindrical can is used to fill a represented by π r 2 h, multiplying the height by x container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the makes the volume x times greater. Multiplying the container if it is each of the following shapes. radius by x makes the volume x 2 times greater, assuming x > 1. a. rectangular prism

42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular b. square prism prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your c. triangular prism with a right triangle as the base reasoning.

SOLUTION: SOLUTION: Francisco; Valerie incorrectly used as the 3 length of one side of the triangular base. Francisco The volume of the can is 20π in . It takes three full used a different approach, but his solution is correct. cans to fill the container, so the volume of the 3 container is 60π in . Francisco used the standard formula for the volume

of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the a. Choose some basic values for 2 of the sides, and base. then determine the third side. Base: 3 by 5.

ANSWER:

Francisco; Valerie incorrectly used as the 3 by 5 by 4π length of one side of the triangular base. Francisco

used a different approach, but his solution is correct. b. Choose some basic values for 2 of the sides, and 43. CHALLENGE A cylindrical can is used to fill a then determine the third side. Base: 5 by 5. container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism

c. triangular prism with a right triangle as the base 5 by 5 by

c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4.

SOLUTION:

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the container is 60π in 3.

3 by 4 by 10π a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. ANSWER: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by

3 by 5 by 4π c. base with legs measuring 3 in. and 4 in., height

10π in. b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION:

Sample answer: The nursery means a cubic yard, 3 5 by 5 by which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine c. Choose some basic values for 2 of the sides, and the number of cubic yards of soil needed. then determine the third side. Base: Legs: 3 by 4. ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters.

3 by 4 by 10π SOLUTION: ANSWER: Choose 3 values that multiply to make 50. The Sample answers: factors of 50 are 2, 5, 5, so these are the simplest values to choose.

a. π 3 by 5 by 4 Sample answer:

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to ANSWER: the following question posted on an Internet Sample answer: gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of

soil? How do I know what to order? SOLUTION: 46. REASONING Determine whether the following statement is true or false. Explain. Sample answer: The nursery means a cubic yard, 3 Two cylinders with the same height and the same which is 3 or 27 cubic feet. Find the volume of your lateral area must have the same volume. garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. SOLUTION: True; if two cylinders have the same height (h = h ) ANSWER: 1 2 Sample answer: The nursery means a cubic yard, and the same lateral area (L1 = L2), the circular 3 which is 3 or 27 cubic feet. Find the volume of your bases must have the same area. garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

Sample answer: ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 same area. Therefore, πr h is the same for each cylinder.

47. WRITING IN MATH How are the formulas for ANSWER: the volume of a prism and the volume of a cylinder

Sample answer: similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2. 46. REASONING Determine whether the following statement is true or false. Explain. ANSWER: Two cylinders with the same height and the same Sample answer: Both formulas involve multiplying the lateral area must have the same volume. area of the base by the height. The base of a prism is a polygon, so the expression representing the area SOLUTION: varies, depending on the type of polygon it is. The True; if two cylinders have the same height (h = h ) 2 1 2 base of a cylinder is a circle, so its area is πr . and the same lateral area (L1 = L2), the circular bases must have the same area. 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm C 17 cm D 11.5 cm

SOLUTION: The radii must also be equal. ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 same area. Therefore, πr h is the same for each cylinder.

47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? ANSWER: SOLUTION: B Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, 49. A cylindrical tank used for oil storage has a height so the expression representing the area varies, that is half the length of its radius. If the volume of depending on the type of polygon it is. The base of a the tank is 1,122,360 ft3, what is the tank’s radius? 2 cylinder is a circle, so its area is πr . F 89.4 ft G 178.8 ft ANSWER: H 280.9 ft Sample answer: Both formulas involve multiplying the J area of the base by the height. The base of a prism is 561.8 ft a polygon, so the expression representing the area SOLUTION: varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr .

ANSWER: F

50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square?

ANSWER: SOLUTION: B The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can 49. A cylindrical tank used for oil storage has a height be written as shown. that is half the length of its radius. If the volume of

the tank is 1,122,360 ft3, what is the tank’s radius? F 89.4 ft G 178.8 ft ANSWER: H 280.9 ft J 561.8 ft SOLUTION: 51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 D $4250 E $42,500 SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C. ANSWER: F ANSWER: C 50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square? Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

SOLUTION: The radius of the circle is 2x and the length of each 52. side of the square is 4x. So, the ratio of the areas can be written as shown. SOLUTION: The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of ANSWER: the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the 51. SAT/ACT A county proposes to enact a new 0.5% Pythagorean Theorem to find the slant height. property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 D $4250 E $42,500 SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary. Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle.

52. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

The perimeter is P = 3 × 10 or 30 feet.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle.

So, the lateral area of the pyramid is about 212.1 ft2.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 2 2 212.1 ft ; 255.4 ft The perimeter is P = 3 × 10 or 30 feet.

53.

SOLUTION: So, the area of the base B is ft2. The lateral area L of a regular pyramid is ,

Find the lateral area L and surface area S of the where is the slant height and P is the perimeter of regular pyramid. the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the lateral area of the pyramid is about 212.1 ft2. 2 So, the lateral area of the pyramid is 126 cm .

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 2 2 212.1 ft ; 255.4 ft Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

53. SOLUTION:

The lateral area L of a regular pyramid is , 54. where is the slant height and P is the perimeter of SOLUTION: the base. The pyramid has a slant height of 15 inches and the Here, the base is a square of side 7 cm and the slant base is a hexagon with sides of 10.5 inches. height is 9 cm. A central angle of the hexagon is or 60°, so the

angle formed in the triangle below is 30°.

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base. Use a trigonometric ratio to find the measure of the apothem a.

Therefore, the surface area of the pyramid is 175 cm2. Find the lateral area and surface area of the pyramid. ANSWER: 2 2 126 cm ; 175 cm

So, the lateral area of the pyramid is 472.5 in2. 54. SOLUTION: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special Use a trigonometric ratio to find the measure of the nonstick coating. A rectangular cake pan is 9 inches apothem a. by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

SOLUTION: Find the lateral area and surface area of the The area that needs to be coated is the sum of the pyramid. lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

So, the lateral area of the pyramid is 472.5 in2. Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2 The diameter of the circle is about 8.3 m. ANSWER: 55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches 8.3 m by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated? 57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER:

2 205 in The diameter of the circle is about 11.4 m.

Find the indicated measure. Round to the ANSWER: nearest tenth. 11.4 cm 56. The area of a circle is 54 square meters. Find the diameter. 58. The area of a circle is 191 square feet. Find the

SOLUTION: radius. SOLUTION:

The diameter of the circle is about 8.3 m.

ANSWER: ANSWER: 8.3 m 7.8 ft

57. Find the diameter of a circle with an area of 102 59. Find the radius of a circle with an area of 271 square

square centimeters. inches. SOLUTION: SOLUTION:

The diameter of the circle is about 11.4 m.

ANSWER: ANSWER: 11.4 cm 9.3 in. 58. The area of a circle is 191 square feet. Find the Find the area of each trapezoid, rhombus, or radius. kite. SOLUTION:

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches.

SOLUTION: ANSWER: 2 120 in

61. SOLUTION: ANSWER: The area A of a trapezoid is one half the product of 9.3 in. the height h and the sum of the lengths of its bases, b1 and b2. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: ANSWER: 2 The area A of a kite is one half the product of the 378 m lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

Find the volume of each prism.

SOLUTION: 1. If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and The volume V of a prism is V = Bh, where B is the an oblique one of the same height and cross sectional area of a base and h is the height of the prism. area are same.

3 The volume is 108 cm . ANSWER: 3 ANSWER: 26.95 m 3 108 cm 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional

2. area are same.

SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 3 396 in 5.

3. the oblique rectangular prism shown at the right SOLUTION:

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and 206.4 ft3 an oblique one of the same height and cross sectional area are same.

6. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right cylinder 26.95 m3 and an oblique one of the same height and cross sectional area are same. 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters

SOLUTION:

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 5. 7.4 inches SOLUTION: SOLUTION:

ANSWER: ANSWER: 3 410.1 in 206.4 ft3 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? 6. A 4000 SOLUTION: B 6400 C If two solids have the same height h and the same 30,000 cross-sectional area B at every level, then they have D 48,000 the same volume. So, the volume of a right cylinder SOLUTION: and an oblique one of the same height and cross sectional area are same.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. ANSWER: 3 ANSWER: 1357.2 m D 7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters CCSS SENSE-MAKING Find the volume of each prism. SOLUTION:

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION: ANSWER: 30 in3

ANSWER: 3 410.1 in 11. SOLUTION: 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to The base is a triangle with a base length of 11 m and be filled to four feet deep and each cubic foot holds the corresponding height of 7 m. The height of the 7.5 gallons, how many gallons will it take to fill the prism is 14 m. lap pool? A 4000 B 6400 C 30,000 D 48,000 SOLUTION: ANSWER: 3 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5)

= 48,000. 12.

Therefore, the correct choice is D. SOLUTION: The base is a right triangle with a leg length of 9 cm ANSWER: and the hypotenuse of length 15 cm. D Use the Pythagorean Theorem to find the height of CCSS SENSE-MAKING Find the volume of the base. each prism.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

The height of the prism is 6 cm.

ANSWER: 3 30 in ANSWER: 324 cm3

11.

SOLUTION: 13. The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the SOLUTION: prism is 14 m. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the

ANSWER: area of a base and h is the height of the prism.

3 2 539 m B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

If two solids have the same height h and the same Use the Pythagorean Theorem to find the height of cross-sectional area B at every level, then they have

the base. the same volume. So, the volume of a right prism and

an oblique one of the same height and cross sectional area are same.

ANSWER: 3 The height of the prism is 6 cm. 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 324 cm3 the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

13.

SOLUTION: If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 1534.25 in3 the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional CCSS SENSE-MAKING Find the volume of area are same. each cylinder. Round to the nearest tenth.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 16. B = 11.4 ft and h = 5.1 ft. Therefore, the volume is SOLUTION: r = 5 yd and h = 18 yd ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters ANSWER: SOLUTION: 3 1413.7 yd If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches ANSWER: SOLUTION: 407.2 cm3 If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 18. SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of

ANSWER: the cylinder.

1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. SOLUTION:

r = 5 yd and h = 18 yd Now you can find the volume.

ANSWER: 3 1413.7 yd ANSWER: 3 823.0 in

17. SOLUTION: r = 6 cm and h = 3.6 cm. 19.

ANSWER: r = 7.5 mm and h = 15.2 mm. 407.2 cm3

18. ANSWER: SOLUTION: 3 r = 5.5 in. 2686.1 mm

Use the Pythagorean Theorem to find the height of 20. PLANTER A planter is in the shape of a the cylinder. rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER: 3 2740.5 in ANSWER: 3 823.0 in 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box? 19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. SOLUTION:

r = 7.5 mm and h = 15.2 mm. The volume of the empty space is the difference of

volumes of the rectangular prism and the cylinders.

ANSWER: 2686.1 mm3 ANSWER: 20. PLANTER A planter is in the shape of a 521.5 cm3 rectangular prism 18 inches long, inches deep, 22. SANDCASTLES In a sandcastle competition, and 12 inches high. What is the volume of potting soil contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct in the planter if the planter is filled to inches amount of sand, they want to create cylinders that below the top? are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION: SOLUTION: The planter is to be filled inches below the top, so 3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: Therefore, the diameter of the cylinders should be 3 2740.5 in about 2.52 ft. ANSWER: 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two 2.52 ft cylindrical candles. Each candle has a diameter of 9 Find the volume of the solid formed by each centimeters and a height of 15 centimeters, as shown net. at the right. What is the volume of the empty space in the box?

23.

SOLUTION: SOLUTION: The middle piece of the net is the front of the solid. The volume of the empty space is the difference of The top and bottom pieces are the bases and the volumes of the rectangular prism and the cylinders. pieces on the ends are the side faces. This is a

triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

ANSWER: The height of the prism is 20 cm.

3 521.5 cm The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism. SANDCASTLES 22. In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION: ANSWER: 3 3 3934.9 cm V = 10 ft and h = 2 ft Use the formula to find r.

Therefore, the diameter of the cylinders should be 24. about 2.52 ft. SOLUTION: ANSWER: The circular bases at the top and bottom of the net 2.52 ft indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. Find the volume of the solid formed by each However, since the middle piece is a parallelogram, it net. is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

If two solids have the same height h and the same cross-sectional area B at every level, then they have 23. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional SOLUTION: area are same. The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

ANSWER: 3 48.9 m The height of the prism is 20 cm. 25. FOOD A cylindrical can of baked potato chips has a The volume V of a prism is V = Bh, where B is the height of 27 centimeters and a radius of 4 area of the base, h is the height of the prism. centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

ANSWER: 3 3934.9 cm SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

24. SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. The height of the new can will be 35.1 cm. However, since the middle piece is a parallelogram, it is oblique. ANSWER: 35.1 cm The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m. 26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. If two solids have the same height h and the same Describe how each change affects the volume of the cross-sectional area B at every level, then they have cylinder. the same volume. So, the volume of a right prism and a. The height is tripled. an oblique one of the same height and cross sectional b. The radius is tripled. area are same. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

ANSWER: 12-4 Volumes of Prisms and Cylinders a. 3 48.9 m When the height is tripled, h = 3h.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a When the height and the radius are tripled, the radius of 5 centimeters and a height of 8 centimeters. volume is multiplied by 27. Describe how each change affects the volume of the cylinder. a. The height is tripled. d. When the dimensions are exchanged, r = 8 and h = 5 cm. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

Compare to the original volume.

a. When the height is tripled, h = 3h. The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. eSolutions Manual - Powered by Cognero Page 8 b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by When the height is tripled, the volume is multiplied by 3. 27. SOIL A soil scientist wants to determine the bulk b. When the radius is tripled, r = 3r. density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a So, when the radius is tripled, the volume is multiplied plant grow in this soil if a bulk density of 0.018 pound by 9. per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its c. When the height and the radius are tripled, r = 3r weight in pounds? and h = 3h. SOLUTION:

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus: Compare to the original volume.

The volume is multiplied by . b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant ANSWER: should grow fairly well.

a. The volume is multiplied by 3. c. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk

density of a potting soil to assess how well a specific plant will grow in it. The density ANSWER: of the soil sample is the ratio of its weight to its 3 volume. a. 0.0019 lb / in

a. b. The plant should grow well in this soil since the If the weight of the container with the soil is 20 3 pounds and the weight of the container alone is 5 bulk density of 0.0019 lb / in is close to the desired 3 pounds, what is the soil’s bulk density? b. Assuming bulk density of 0.0018 lb / in . that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound c. 8.3 lb per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its Find the volume of each composite solid. Round weight in pounds? to the nearest tenth if necessary. SOLUTION:

28.

SOLUTION: a. First calculate the volume of soil in the pot. Then The solid is a combination of two rectangular prisms. divide the weight of the soil by the volume. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

The weight of the soil is the weight of the pot with

soil minus the weight of the pot. W = 20 – 5 = 15 lbs. ANSWER: 3 The soil density is thus: 225 cm

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

c. 29.

SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb ANSWER: Find the volume of each composite solid. Round 3 120 m to the nearest tenth if necessary.

and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: ANSWER: 225 cm3 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder? 29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a

radius of and a height of 11.5 cm. The empty space used to keep the can has a radius

of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 ANSWER: volume V of a cylinder is V = πr h, where r is the 3 radius of the base and h is the height of the cylinder. 120 m

Therefore, the volume of the rubberized material is 3 30. about 304.1 cm . SOLUTION: ANSWER: The solid is a combination of a rectangular prism and 3 304.1 cm two half cylinders. Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the ANSWER: volume of the rubberized material that makes up the 5.7 cm holder? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 ANSWER: centimeters. What is the height? 678.6 in3 SOLUTION: 34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches Find the volume. and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

SOLUTION: The radius is 6. Find the volume. The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? ANSWER: 3 SOLUTION: 3,190,680.0 cm Use the surface area formula to find the length of the 36. CCSS MODELING The base of a rectangular base of the prism. swimming pool is sloped so one end of the pool is 6

feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

Find the volume. SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two ANSWER: prisms. 3 576 in

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. ANSWER: 3 SOLUTION: 1575 ft The volume will be the highest when the diameter is CHANGING DIMENSIONS 95 cm and will be the lowest when it is 30 cm.That is 37. A soy milk company when the radii are 47.5 cm and 15 cm respectively. is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The Find the difference between the volumes. company wants the base of the container to stay the same. What will be the height of the new containers?

SOLUTION: Find the volume of the original container. ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown The volume of the new container is 125% of the in the figure. If the width is 15 feet, find the volume original container, with the same base dimensions. of water it takes to fill the pool. Use 1.25V and B to find h.

SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ANSWER: ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two 38. DESIGN Sketch and label (in inches) three different prisms. designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

ANSWER: 3 1575 ft

37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the The last equation gives us a relation between the same. What will be the height of the new containers? radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height.

SOLUTION:

Find the volume of the original container. If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

For any rectangular container, the volume equation is: ANSWER:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = DESIGN 38. Sketch and label (in inches) three different 2.5 in. designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius. ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 ANSWER: 1100 cm ; Each triangular prism has a base area of Sample answers: 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option. SOLUTION: ANSWER: The base of the prism can be divided into 5 Because 2.96 yd3 of concrete are needed, the congruent triangles of a base 8 cm and the second contractor is less expensive at $2181.50. corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 41. MULTIPLE REPRESENTATIONS In this 10 cm. Find the base area of each triangular prism. problem, you will investigate right and oblique cylinders . Therefore, the volume of the pentagonal prism is a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters. ANSWER: 3 1100 cm ; Each triangular prism has a base area of b. VERBAL A square prism has a height of 10 2 meters and a base edge of 6 meters. Is its volume or 22 cm and a height of 10 cm. greater than, less than, or equal to the volume of the cylinder? Explain. 40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 c. ANALYTICAL Describe which change affects feet long, 12 feet wide, and 4 inches thick. One the volume of the cylinder more: multiplying the contractor bid $2225 for the project. A second height by x or multiplying the radius by x. Explain. contractor bid $500 per cubic yard for the new patio SOLUTION: and $700 for removal of the old patio. Which is the less expensive option? Explain. a. The oblique cylinder should look like the right cylinder (same height and size), except that it is SOLUTION: pushed a little to the side, like a slinky. Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume. b. Find the volume of each.

The total cost for the second contractor is about . The volume of the square prism is greater. Therefore, the second contractor is a less expensive c. Do each scenario. option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

the cylinder? Explain. than x. ANSWER: c. ANALYTICAL Describe which change affects a. the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater. b. Find the volume of each. c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

The volume of the square prism is greater. c. Do each scenario.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

2 For example, if x = 0.5, then x = 0.25, which is less ANSWER: than x. Francisco; Valerie incorrectly used as the ANSWER: length of one side of the triangular base. Francisco a. used a different approach, but his solution is correct. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. Greater than; a square with a side length of 6 m b. square prism 2 has an area of 36 m . A circle with a diameter of 6 2 c. triangular prism with a right triangle as the base m has an area of 9π or 28.3 m . Since the heights are the same, the volume of the square prism is greater.

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the SOLUTION: radius by x makes the volume x 2 times greater, assuming x > 1. 3 The volume of the can is 20π in . It takes three full 42. CCSS CRITIQUE Franciso and Valerie each cans to fill the container, so the volume of the calculated the volume of an equilateral triangular container is 60π in 3. prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your a. Choose some basic values for 2 of the sides, and reasoning. then determine the third side. Base: 3 by 5.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco 3 by 5 by 4π used a different approach, but his solution is correct. b. Choose some basic values for 2 of the sides, and Francisco used the standard formula for the volume then determine the third side. Base: 5 by 5. of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

ANSWER:

Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco 5 by 5 by used a different approach, but his solution is correct. c. Choose some basic values for 2 of the sides, and 43. CHALLENGE A cylindrical can is used to fill a then determine the third side. Base: Legs: 3 by 4. container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism

c. triangular prism with a right triangle as the base 3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π

SOLUTION: b. 5 by 5 by

3 The volume of the can is 20π in . It takes three full c. base with legs measuring 3 in. and 4 in., height cans to fill the container, so the volume of the 10π in. container is 60π in 3. 44. WRITING IN MATH Write a helpful response to a. Choose some basic values for 2 of the sides, and the following question posted on an Internet then determine the third side. Base: 3 by 5. gardening forum.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters.

SOLUTION: 5 by 5 by Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest c. Choose some basic values for 2 of the sides, and values to choose. then determine the third side. Base: Legs: 3 by 4. Sample answer:

ANSWER: 3 by 4 by 10π Sample answer: ANSWER: Sample answers:

a. 3 by 5 by 4π

46. REASONING Determine whether the following b. 5 by 5 by statement is true or false. Explain. Two cylinders with the same height and the same c. base with legs measuring 3 in. and 4 in., height lateral area must have the same volume. 10π in. SOLUTION: 44. WRITING IN MATH Write a helpful response to True; if two cylinders have the same height (h1 = h2) the following question posted on an Internet gardening forum. and the same lateral area (L1 = L2), the circular bases must have the same area. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. The radii must also be equal.

ANSWER: ANSWER: Sample answer: The nursery means a cubic yard, True; if two cylinders have the same height and the 3 same lateral area, the circular bases must have the which is 3 or 27 cubic feet. Find the volume of your 2 garden in cubic feet and divide by 27 to determine same area. Therefore, πr h is the same for each the number of cubic yards of soil needed. cylinder.

45. OPEN ENDED Draw and label a prism that has a 47. WRITING IN MATH How are the formulas for volume of 50 cubic centimeters. the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Choose 3 values that multiply to make 50. The SOLUTION: factors of 50 are 2, 5, 5, so these are the simplest Both formulas involve multiplying the area of the values to choose. base by the height. The base of a prism is a polygon, so the expression representing the area varies, Sample answer: depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area ANSWER: varies, depending on the type of polygon it is. The 2 Sample answer: base of a cylinder is a circle, so its area is πr . 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm 46. REASONING Determine whether the following C statement is true or false. Explain. 17 cm Two cylinders with the same height and the same D 11.5 cm lateral area must have the same volume. SOLUTION: SOLUTION:

True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: B

The radii must also be equal. 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of ANSWER: the tank is 1,122,360 ft3, what is the tank’s radius? True; if two cylinders have the same height and the F 89.4 ft same lateral area, the circular bases must have the 2 G 178.8 ft same area. Therefore, πr h is the same for each H 280.9 ft cylinder. J 561.8 ft 47. WRITING IN MATH How are the formulas for SOLUTION: the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr . ANSWER: F 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs 50. SHORT RESPONSE What is the ratio of the area measuring 8 centimeters and 15 centimeters. What is of the circle to the area of the square? the height of the prism? A 34.5 cm B 23 cm C 17 cm D 11.5 cm SOLUTION: SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax ANSWER: amount for a landowner whose property has a

B taxable value of $85,000? A $4.25 49. A cylindrical tank used for oil storage has a height B $170 that is half the length of its radius. If the volume of C $425 the tank is 1,122,360 ft3, what is the tank’s radius? D $4250 F 89.4 ft E $42,500 G 178.8 ft SOLUTION: H 280.9 ft Find the 0.5% of $85,000. J 561.8 ft SOLUTION: Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

ANSWER: The lateral area L of a regular pyramid is , F where is the slant height and P is the perimeter of 50. SHORT RESPONSE What is the ratio of the area the base. of the circle to the area of the square? The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 D $4250 E $42,500 Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem SOLUTION: to find the height h of the triangle. Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , The perimeter is P = 3 × 10 or 30 feet.

where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. Therefore, the surface area of the pyramid is about 2 255.4 ft . ANSWER: 212.1 ft2; 255.4 ft2

53.

SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant The perimeter is P = 3 × 10 or 30 feet. height is 9 cm.

So, the lateral area of the pyramid is 126 cm2. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the The surface area S of a regular regular pyramid. pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is about 212.1 ft2. Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 2 255.4 ft . 54. ANSWER: SOLUTION: 212.1 ft2; 255.4 ft2 The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

53. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant Use a trigonometric ratio to find the measure of the height is 9 cm. apothem a.

So, the lateral area of the pyramid is 126 cm2. Find the lateral area and surface area of the pyramid. The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 54. 472.5 in2; 758.9 in2 SOLUTION: The pyramid has a slant height of 15 inches and the 55. BAKING Many baking pans are given a special base is a hexagon with sides of 10.5 inches. nonstick coating. A rectangular cake pan is 9 inches A central angle of the hexagon is or 60°, so the by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated? angle formed in the triangle below is 30°.

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

Use a trigonometric ratio to find the measure of the ANSWER: apothem a. 2 205 in Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter.

SOLUTION: Find the lateral area and surface area of the pyramid.

So, the lateral area of the pyramid is 472.5 in2.

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION: Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated? The diameter of the circle is about 11.4 m. ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the SOLUTION: radius. The area that needs to be coated is the sum of the SOLUTION: lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. ANSWER: SOLUTION: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches. SOLUTION:

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. ANSWER: SOLUTION: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60.

SOLUTION: The diameter of the circle is about 11.4 m. The area A of a kite is one half the product of the lengths of its diagonals, d and d . ANSWER: 1 2

11.4 cm d1 = 12 in. and d2 = 7 + 13 = 20 in. 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 2 120 in

ANSWER: 7.8 ft 61. 59. Find the radius of a circle with an area of 271 square SOLUTION: inches. The area A of a trapezoid is one half the product of SOLUTION: the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

Find the volume of each prism.

SOLUTION: If two solids have the same height h and the same 1. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and SOLUTION: an oblique one of the same height and cross sectional The volume V of a prism is V = Bh, where B is the area are same. area of a base and h is the height of the prism.

3 The volume is 108 cm . ANSWER: 26.95 m3

ANSWER: 4. an oblique pentagonal prism with a base area of 42 108 cm3 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

2. SOLUTION: The volume V of a prism is V = Bh, where B is the

area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 5. 3 396 in SOLUTION:

3. the oblique rectangular prism shown at the right

SOLUTION: ANSWER: If two solids have the same height h and the same 3 cross-sectional area B at every level, then they have 206.4 ft the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

6. SOLUTION: If two solids have the same height h and the same

cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder ANSWER: and an oblique one of the same height and cross

3 sectional area are same. 26.95 m 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 3 area are same. 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION:

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the ANSWER: nearest tenth. 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches 5. SOLUTION: SOLUTION:

ANSWER: 3 410.1 in ANSWER: 206.4 ft3 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 6. B 6400 C 30,000 SOLUTION: D 48,000 If two solids have the same height h and the same cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: ANSWER: 3 D 1357.2 m CCSS SENSE-MAKING Find the volume of 7. a cylinder with a diameter of 16 centimeters and a each prism. height of 5.1 centimeters SOLUTION: 10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches ANSWER: SOLUTION: 30 in3

ANSWER: 11. 3 410.1 in SOLUTION: The base is a triangle with a base length of 11 m and 9. MULTIPLE CHOICE A rectangular lap pool the corresponding height of 7 m. The height of the measures 80 feet long by 20 feet wide. If it needs to prism is 14 m. be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000 D 48,000 ANSWER: 3 SOLUTION: 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) 12. = 48,000. SOLUTION:

Therefore, the correct choice is D. The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. ANSWER: D Use the Pythagorean Theorem to find the height of the base. CCSS SENSE-MAKING Find the volume of each prism.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. The height of the prism is 6 cm.

ANSWER: ANSWER: 3 3 30 in 324 cm

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is 3 539 m

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 12. centimeters and with a base area of 136 square centimeters SOLUTION: The base is a right triangle with a leg length of 9 cm SOLUTION: and the hypotenuse of length 15 cm. If two solids have the same height h and the same cross-sectional area B at every level, then they have Use the Pythagorean Theorem to find the height of the same volume. So, the volume of a right prism and the base. an oblique one of the same height and cross sectional area are same.

ANSWER: 3 2040 cm

The height of the prism is 6 cm. 15. a square prism with a base edge of 9.5 inches and a

height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 324 cm3 area are same.

13. ANSWER: SOLUTION: 3 If two solids have the same height h and the same 1534.25 in cross-sectional area B at every level, then they have CCSS SENSE-MAKING Find the volume of the same volume. So, the volume of a right prism and each cylinder. Round to the nearest tenth. an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 16.

2 SOLUTION: B = 11.4 ft and h = 5.1 ft. Therefore, the volume is r = 5 yd and h = 18 yd

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square ANSWER: centimeters 3 1413.7 yd SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 17. area are same. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a ANSWER: height of 17 inches 3 407.2 cm SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 18. SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

ANSWER: 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16.

SOLUTION: r = 5 yd and h = 18 yd Now you can find the volume.

ANSWER: 3 ANSWER: 1413.7 yd 3 823.0 in

17. 19. SOLUTION: r = 6 cm and h = 3.6 cm. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm. ANSWER: 407.2 cm3

18. ANSWER: 2686.1 mm3 SOLUTION: r = 5.5 in. 20. PLANTER A planter is in the shape of a

Use the Pythagorean Theorem to find the height of rectangular prism 18 inches long, inches deep, the cylinder. and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER: 3 2740.5 in

ANSWER: 21. SHIPPING A box 18 centimeters by 9 centimeters 3 by 15 centimeters is being used to ship two 823.0 in cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box?

ANSWER: ANSWER: 2686.1 mm3 521.5 cm3 20. PLANTER A planter is in the shape of a 22. SANDCASTLES In a sandcastle competition, rectangular prism 18 inches long, inches deep, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct and 12 inches high. What is the volume of potting soil amount of sand, they want to create cylinders that in the planter if the planter is filled to inches are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? below the top? SOLUTION: SOLUTION: 3 The planter is to be filled inches below the top, so V = 10 ft and h = 2 ft Use the formula to find r.

Therefore, the diameter of the cylinders should be ANSWER: about 2.52 ft. 3 2740.5 in ANSWER: 2.52 ft 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two Find the volume of the solid formed by each cylindrical candles. Each candle has a diameter of 9 net. centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box?

23. SOLUTION:

The middle piece of the net is the front of the solid. SOLUTION: The top and bottom pieces are the bases and the The volume of the empty space is the difference of pieces on the ends are the side faces. This is a volumes of the rectangular prism and the cylinders. triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm.

ANSWER: The volume V of a prism is V = Bh, where B is the 521.5 cm3 area of the base, h is the height of the prism.

3 V = 10 ft and h = 2 ft Use the formula to find r.

Find the volume of the solid formed by each is oblique.

net. The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 23. area are same. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base. ANSWER: 3 48.9 m

The height of the prism is 20 cm. 25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% The volume V of a prism is V = Bh, where B is the larger than the regular can. If both cans have the area of the base, h is the height of the prism. same radius, what is the height of the larger can?

ANSWER: SOLUTION: 3934.9 cm3 The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

24. SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece The height of the new can will be 35.1 cm. were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it ANSWER: is oblique. 35.1 cm

The radius is 1.8 m, the height is 4.8 m, and the slant 26. CHANGING DIMENSIONS A cylinder has a height is 6 m. radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. If two solids have the same height h and the same a. cross-sectional area B at every level, then they have The height is tripled. the same volume. So, the volume of a right prism and b. The radius is tripled. an oblique one of the same height and cross sectional c. Both the radius and the height are tripled. area are same. d. The dimensions are exchanged.

SOLUTION:

a. ANSWER: When the height is tripled, h = 3h. 3 48.9 m

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm When the height and the radius are tripled, the 26. CHANGING DIMENSIONS A cylinder has a volume is multiplied by 27. radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the d. When the dimensions are exchanged, r = 8 and h cylinder. = 5 cm. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

Compare to the original volume.

The volume is multiplied by . a. When the height is tripled, h = 3h. ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

When the height is tripled, the volume is multiplied by 27. SOIL A soil scientist wants to determine the bulk 3. density of a potting soil to assess how well a specific b. When the radius is tripled, r = 3r. plant will grow in it. The density of the soil sample is the ratio of its weight to its volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. So, when the radius is tripled, the volume is multiplied c. If a bag of this soil holds 2.5 cubic feet, what is its by 9. weight in pounds?

c. When the height and the radius are tripled, r = 3r SOLUTION: and h = 3h.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus:

Compare to the original volume.

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant The volume is multiplied by . 12-4 Volumes of Prisms and Cylinders should grow fairly well.

ANSWER: c. a. The volume is multiplied by 3. 2 b. The volume is multiplied by 3 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific ANSWER: 3 plant will grow in it. The density a. 0.0019 lb / in of the soil sample is the ratio of its weight to its volume. b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired a. If the weight of the container with the soil is 20 3 pounds and the weight of the container alone is 5 bulk density of 0.0018 lb / in . pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a c. 8.3 lb plant grow in this soil if a bulk density of 0.018 pound Find the volume of each composite solid. Round per square inch is desirable for root growth? Explain. to the nearest tenth if necessary. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION:

28. SOLUTION: The solid is a combination of two rectangular prisms.

The base of one rectangular prism is 5 cm by 3 cm a. First calculate the volume of soil in the pot. Then and the height is 11 cm. The base of the other prism divide the weight of the soil by the volume. is 4 cm by 3 cm and the height is 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. ANSWER: W = 20 – 5 = 15 lbs. 225 cm3

The soil density is thus:

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. 29. c. SOLUTION: eSolutions Manual - Powered by Cognero The solid is a combination of a rectangular prismPage and 9

a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in . ANSWER: c. 8.3 lb 3 120 m Find the volume of each composite solid. Round to the nearest tenth if necessary.

30.

28. SOLUTION: The solid is a combination of a rectangular prism and SOLUTION: two half cylinders. The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder. ANSWER:

3 120 m

Therefore, the volume of the rubberized material is about 304.1 cm3.

30. ANSWER: 3 SOLUTION: 304.1 cm The solid is a combination of a rectangular prism and two half cylinders. Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the ANSWER: thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the 5.7 cm volume of the rubberized material that makes up the holder? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic ANSWER: centimeters. The diameter of the can is 9 678.6 in3 centimeters. What is the height? 34. A rectangular prism has a surface area of 432 SOLUTION: square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER: 5.7 cm Find the volume.

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

Find the difference between the volumes.

ANSWER: 3 678.6 in

34. A rectangular prism has a surface area of 432 ANSWER: square inches, a height of 6 inches, and a width of 12 3 inches. What is the volume? 3,190,680.0 cm

SOLUTION: 36. CCSS MODELING The base of a rectangular Use the surface area formula to find the length of the swimming pool is sloped so one end of the pool is 6 base of the prism. feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

SOLUTION: Find the volume. The swimming pool is a combination of a rectangular

prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two

prisms. ANSWER: 3 576 in

SOLUTION: 37. CHANGING DIMENSIONS A soy milk company The volume will be the highest when the diameter is is planning a promotion in which the volume of soy 95 cm and will be the lowest when it is 30 cm.That is milk in each container will be increased by 25%. The when the radii are 47.5 cm and 15 cm respectively. company wants the base of the container to stay the same. What will be the height of the new containers? Find the difference between the volumes.

SOLUTION:

Find the volume of the original container.

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 The volume of the new container is 125% of the feet deep and the other end is 3 feet deep, as shown original container, with the same base dimensions. in the figure. If the width is 15 feet, find the volume Use 1.25V and B to find h. of water it takes to fill the pool.

prisms. 3 drawing. (1 cup ≈ 14.4375 in ) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

ANSWER: 3 1575 ft

37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The The last equation gives us a relation between the company wants the base of the container to stay the radius and height of the cylinder that must be fulfilled same. What will be the height of the new containers? to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height.

SOLUTION: Find the volume of the original container. If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

For any rectangular container, the volume equation is:

Sample answers:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius. ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of ANSWER: 2 or 22 cm and a height of 10 cm. Sample answers:

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by

dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism. The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option.

ANSWER: SOLUTION: Because 2.96 yd3 of concrete are needed, the The base of the prism can be divided into 5 second contractor is less expensive at $2181.50. congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal 41. MULTIPLE REPRESENTATIONS In this prism is a combination of 5 triangular prisms of height problem, you will investigate right and oblique 10 cm. Find the base area of each triangular prism. cylinders . a. GEOMETRIC Draw a right cylinder and an Therefore, the volume of the pentagonal prism is oblique cylinder with a height of 10 meters and a diameter of 6 meters.

ANSWER: b. VERBAL A square prism has a height of 10 3 meters and a base edge of 6 meters. Is its volume 1100 cm ; Each triangular prism has a base area of greater than, less than, or equal to the volume of 2 or 22 cm and a height of 10 cm. the cylinder? Explain.

c. ANALYTICAL Describe which change affects 40. PATIOS Mr. Thomas is planning to remove an old the volume of the cylinder more: multiplying the patio and install a new rectangular concrete patio 20 height by x or multiplying the radius by x. Explain. feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second SOLUTION: contractor bid $500 per cubic yard for the new patio a. The oblique cylinder should look like the right and $700 for removal of the old patio. Which is the cylinder (same height and size), except that it is less expensive option? Explain. pushed a little to the side, like a slinky. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd b. Find the volume of each.

Find the volume.

The total cost for the second contractor is about The volume of the square prism is greater. . c. Do each scenario.

Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

b. VERBAL A square prism has a height of 10 For example, if x = 0.5, then x2 = 0.25, which is less meters and a base edge of 6 meters. Is its volume than x. greater than, less than, or equal to the volume of the cylinder? Explain. ANSWER: a. c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky. b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater.

b. Find the volume of each. c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater. ANSWER: 2 For example, if x = 0.5, then x = 0.25, which is less Francisco; Valerie incorrectly used as the than x. length of one side of the triangular base. Francisco used a different approach, but his solution is correct. ANSWER: a. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism

b. Greater than; a square with a side length of 6 m 2 c. triangular prism with a right triangle as the base has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater.

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x SOLUTION: makes the volume x times greater. Multiplying the 3 radius by x makes the volume x 2 times greater, The volume of the can is 20π in . It takes three full assuming x > 1. cans to fill the container, so the volume of the container is 60π in 3. 42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 a. Choose some basic values for 2 of the sides, and units. Is either of them correct? Explain your then determine the third side. Base: 3 by 5. reasoning.

SOLUTION: 3 by 5 by 4π Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco b. used a different approach, but his solution is correct. Choose some basic values for 2 of the sides, and

then determine the third side. Base: 5 by 5. Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

ANSWER: 5 by 5 by Francisco; Valerie incorrectly used as the

length of one side of the triangular base. Francisco used a different approach, but his solution is correct. c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism 3 by 4 by 10π

c. triangular prism with a right triangle as the base ANSWER: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by SOLUTION: c. base with legs measuring 3 in. and 4 in., height 3 The volume of the can is 20π in . It takes three full 10π in. cans to fill the container, so the volume of the 3 44. WRITING IN MATH Write a helpful response to container is 60π in . the following question posted on an Internet gardening forum. a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your 3 by 5 by 4π garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

c. Choose some basic values for 2 of the sides, and Sample answer: then determine the third side. Base: Legs: 3 by 4.

ANSWER: Sample answer:

3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π 46. REASONING Determine whether the following statement is true or false. Explain. b. 5 by 5 by Two cylinders with the same height and the same lateral area must have the same volume. c. base with legs measuring 3 in. and 4 in., height SOLUTION: 10π in. True; if two cylinders have the same height (h1 = h2) WRITING IN MATH 44. Write a helpful response to and the same lateral area (L1 = L2), the circular the following question posted on an Internet bases must have the same area. gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your The radii must also be equal. garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. ANSWER: True; if two cylinders have the same height and the ANSWER: same lateral area, the circular bases must have the Sample answer: The nursery means a cubic yard, 2 3 same area. Therefore, πr h is the same for each which is 3 or 27 cubic feet. Find the volume of your cylinder. garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder 45. OPEN ENDED Draw and label a prism that has a similar? How are they different? volume of 50 cubic centimeters. SOLUTION: SOLUTION: Both formulas involve multiplying the area of the Choose 3 values that multiply to make 50. The base by the height. The base of a prism is a polygon, factors of 50 are 2, 5, 5, so these are the simplest so the expression representing the area varies, values to choose. depending on the type of polygon it is. The base of a 2 Sample answer: cylinder is a circle, so its area is πr . ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 ANSWER: base of a cylinder is a circle, so its area is πr . Sample answer: 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm C 17 cm 46. REASONING Determine whether the following D 11.5 cm statement is true or false. Explain. Two cylinders with the same height and the same SOLUTION: lateral area must have the same volume. SOLUTION:

True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: B 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of The radii must also be equal. the tank is 1,122,360 ft3, what is the tank’s radius? ANSWER: F 89.4 ft True; if two cylinders have the same height and the G 178.8 ft same lateral area, the circular bases must have the H 280.9 ft 2 same area. Therefore, πr h is the same for each J 561.8 ft cylinder. SOLUTION: 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The ANSWER: 2 F base of a cylinder is a circle, so its area is πr . 50. SHORT RESPONSE What is the ratio of the area 48. The volume of a triangular prism is 1380 cubic of the circle to the area of the square? centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm

B 23 cm C 17 cm SOLUTION: D 11.5 cm The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can SOLUTION: be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? ANSWER: A $4.25 B B $170 49. A cylindrical tank used for oil storage has a height C $425 that is half the length of its radius. If the volume of D $4250 3 E $42,500 the tank is 1,122,360 ft , what is the tank’s radius? F 89.4 ft SOLUTION: G 178.8 ft Find the 0.5% of $85,000. H 280.9 ft J 561.8 ft Therefore, the correct choice is C. SOLUTION: ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , ANSWER: where is the slant height and P is the perimeter of F the base.

50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square? The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION: The perimeter is P = 3 × 10 or 30 feet. The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

Find the perimeter and area of the equilateral Therefore, the surface area of the pyramid is about triangle for the base. Use the Pythagorean Theorem 2 255.4 ft . to find the height h of the triangle. ANSWER: 212.1 ft2; 255.4 ft2

53.

SOLUTION: The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm. The perimeter is P = 3 × 10 or 30 feet.

So, the lateral area of the pyramid is 126 cm2.

2 So, the area of the base B is ft . The surface area S of a regular pyramid is , where Find the lateral area L and surface area S of the L is the lateral area and B is the area of the base. regular pyramid.

2 Therefore, the surface area of the pyramid is 175 So, the lateral area of the pyramid is about 212.1 ft . 2 cm . ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 54. 2 255.4 ft . SOLUTION: ANSWER: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. 2 2 212.1 ft ; 255.4 ft A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

53. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of Use a trigonometric ratio to find the measure of the the base. apothem a. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the lateral area of the pyramid is 126 cm2. Find the lateral area and surface area of the pyramid.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2 54. SOLUTION: 55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches The pyramid has a slant height of 15 inches and the by 13 inches by 2 inches deep. What is the area of base is a hexagon with sides of 10.5 inches. the inside of the pan that needs to be coated? A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: Use a trigonometric ratio to find the measure of the 205 in2 apothem a. Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

Find the lateral area and surface area of the pyramid.

So, the lateral area of the pyramid is 472.5 in2. The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches The diameter of the circle is about 11.4 m. by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated? ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION: SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 ANSWER: square centimeters. 9.3 in. SOLUTION: Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION:

The area A of a kite is one half the product of the

The diameter of the circle is about 11.4 m. lengths of its diagonals, d1 and d2.

ANSWER: d1 = 12 in. and d2 = 7 + 13 = 20 in. 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 2 120 in

ANSWER: 61. 7.8 ft SOLUTION: 59. Find the radius of a circle with an area of 271 square The area A of a trapezoid is one half the product of inches. the height h and the sum of the lengths of its bases, b and b . SOLUTION: 1 2

ANSWER: 2 378 m

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

Find the volume of each prism.

SOLUTION: 1. If two solids have the same height h and the same cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right prism and The volume V of a prism is V = Bh, where B is the an oblique one of the same height and cross sectional area of a base and h is the height of the prism. area are same.

SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 3 396 in 5. SOLUTION: 3. the oblique rectangular prism shown at the right

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right cylinder 26.95 m3 and an oblique one of the same height and cross sectional area are same. 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION:

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 5. 7.4 inches SOLUTION: SOLUTION:

ANSWER: ANSWER: 3 410.1 in 206.4 ft3 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? 6. A 4000 B 6400 SOLUTION: C 30,000 If two solids have the same height h and the same D cross-sectional area B at every level, then they have 48,000 the same volume. So, the volume of a right cylinder SOLUTION: and an oblique one of the same height and cross sectional area are same.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. ANSWER: 3 ANSWER: 1357.2 m D 7. a cylinder with a diameter of 16 centimeters and a CCSS SENSE-MAKING Find the volume of height of 5.1 centimeters each prism. SOLUTION:

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION: ANSWER: 30 in3

ANSWER: 3 410.1 in 11. SOLUTION: 9. MULTIPLE CHOICE A rectangular lap pool The base is a triangle with a base length of 11 m and measures 80 feet long by 20 feet wide. If it needs to the corresponding height of 7 m. The height of the be filled to four feet deep and each cubic foot holds prism is 14 m. 7.5 gallons, how many gallons will it take to fill the

lap pool? A 4000 B 6400 C 30,000 D 48,000 SOLUTION: ANSWER: 3 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000. 12.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

The height of the prism is 6 cm.

ANSWER: 3 30 in ANSWER: 324 cm3

11. SOLUTION: 13. The base is a triangle with a base length of 11 m and SOLUTION: the corresponding height of 7 m. The height of the prism is 14 m. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. ANSWER:

3 2 539 m B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

ANSWER: 58.14 ft3

12. 14. an oblique hexagonal prism with a height of 15 SOLUTION: centimeters and with a base area of 136 square The base is a right triangle with a leg length of 9 cm centimeters and the hypotenuse of length 15 cm. SOLUTION: If two solids have the same height h and the same Use the Pythagorean Theorem to find the height of cross-sectional area B at every level, then they have the base. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER:

3 The height of the prism is 6 cm. 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches

SOLUTION: If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 324 cm3 the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

13.

SOLUTION: ANSWER: If two solids have the same height h and the same cross-sectional area B at every level, then they have 1534.25 in3 the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional CCSS SENSE-MAKING Find the volume of area are same. each cylinder. Round to the nearest tenth.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 16. 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is SOLUTION: r = 5 yd and h = 18 yd ANSWER: 58.14 ft3

ANSWER: 3 2040 cm

Use the Pythagorean Theorem to find the height of the cylinder. ANSWER:

1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. SOLUTION:

r = 5 yd and h = 18 yd Now you can find the volume.

ANSWER:

3 1413.7 yd ANSWER: 3 823.0 in

17. SOLUTION: r = 6 cm and h = 3.6 cm. 19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: r = 7.5 mm and h = 15.2 mm. 407.2 cm3

18. ANSWER: SOLUTION: 3 r = 5.5 in. 2686.1 mm

Use the Pythagorean Theorem to find the height of 20. PLANTER A planter is in the shape of a the cylinder. rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

area are same. SOLUTION:

The volume of the empty space is the difference of r = 7.5 mm and h = 15.2 mm. volumes of the rectangular prism and the cylinders.

ANSWER: 2686.1 mm3 ANSWER: 20. PLANTER A planter is in the shape of a 521.5 cm3

rectangular prism 18 inches long, inches deep, 22. SANDCASTLES In a sandcastle competition, and 12 inches high. What is the volume of potting soil contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct in the planter if the planter is filled to inches amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. below the top? What should the diameter of the cylinders be? SOLUTION: SOLUTION: The planter is to be filled inches below the top, so 3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: Therefore, the diameter of the cylinders should be 3 about 2.52 ft. 2740.5 in ANSWER: 21. SHIPPING A box 18 centimeters by 9 centimeters 2.52 ft by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 Find the volume of the solid formed by each centimeters and a height of 15 centimeters, as shown net. at the right. What is the volume of the empty space in the box?

23.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

ANSWER: The height of the prism is 20 cm.

3 521.5 cm The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism. 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION: ANSWER: 3934.9 cm3 3 V = 10 ft and h = 2 ft Use the formula to find r.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base. ANSWER: 3 48.9 m The height of the prism is 20 cm. 25. FOOD A cylindrical can of baked potato chips has a The volume V of a prism is V = Bh, where B is the height of 27 centimeters and a radius of 4 area of the base, h is the height of the prism. centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

ANSWER: 3934.9 cm3 SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

ANSWER: a. 3 48.9 m When the height is tripled, h = 3h.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

CHANGING DIMENSIONS 26. A cylinder has a When the height and the radius are tripled, the radius of 5 centimeters and a height of 8 centimeters. volume is multiplied by 27. Describe how each change affects the volume of the cylinder. d. When the dimensions are exchanged, r = 8 and h a. The height is tripled. = 5 cm. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

Compare to the original volume.

a. When the height is tripled, h = 3h. The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

So, when the radius is tripled, the volume is multiplied plant grow in this soil if a bulk density of 0.018 pound by 9. per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? c. When the height and the radius are tripled, r = 3r and h = 3h. SOLUTION:

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus: Compare to the original volume.

The volume is multiplied by . b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. ANSWER:

a. The volume is multiplied by 3. c. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

SOIL 27. A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density ANSWER: of the soil sample is the ratio of its weight to its 3 a. 0.0019 lb / in volume.

b. The plant should grow well in this soil since the a. If the weight of the container with the soil is 20 3 pounds and the weight of the container alone is 5 bulk density of 0.0019 lb / in is close to the desired 3 pounds, what is the soil’s bulk density? b. Assuming bulk density of 0.0018 lb / in . that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound c. 8.3 lb per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its Find the volume of each composite solid. Round weight in pounds? to the nearest tenth if necessary. SOLUTION:

28.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs. ANSWER: 3 225 cm The soil density is thus:

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

c. 29.

SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the 3 bulk density of 0.0019 lb / in is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb ANSWER: 3 Find the volume of each composite solid. Round 120 m to the nearest tenth if necessary.

30. 28. SOLUTION: SOLUTION: The solid is a combination of a rectangular prism and The solid is a combination of two rectangular prisms. two half cylinders. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: ANSWER: 12-4225 Volumes cm3 of Prisms and Cylinders 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder? 29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm.

The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 ANSWER: volume V of a cylinder is V = πr h, where r is the 3 radius of the base and h is the height of the cylinder. 120 m

Therefore, the volume of the rubberized material is 3 30. about 304.1 cm . SOLUTION: ANSWER: 3 The solid is a combination of a rectangular prism and 304.1 cm two half cylinders. Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 eSolutionscentimetersManual - Poweredtall, includingby Cognero 1 centimeter for the Page 10 thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the ANSWER: volume of the rubberized material that makes up the 5.7 cm holder? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 ANSWER: centimeters. What is the height? 3 678.6 in SOLUTION: 34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches Find the volume. and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 678.6 in3

base of the prism. swimming pool is sloped so one end of the pool is 6

feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. ANSWER: 3 SOLUTION: 1575 ft The volume will be the highest when the diameter is 37. CHANGING DIMENSIONS A soy milk company 95 cm and will be the lowest when it is 30 cm.That is is planning a promotion in which the volume of soy when the radii are 47.5 cm and 15 cm respectively. milk in each container will be increased by 25%. The

company wants the base of the container to stay the Find the difference between the volumes. same. What will be the height of the new containers?

SOLUTION: Find the volume of the original container. ANSWER: 3,190,680.0 cm3

SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the ANSWER: rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two 38. DESIGN Sketch and label (in inches) three different prisms. designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following

equation for volume:

ANSWER: 3 1575 ft

SOLUTION:

Find the volume of the original container. If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

For any rectangular container, the volume equation is: ANSWER:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 38. DESIGN Sketch and label (in inches) three different 2.5 in. designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius. ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 ANSWER: 1100 cm ; Each triangular prism has a base area of Sample answers: 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

The total cost for the second contractor is about .

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 c. ANALYTICAL Describe which change affects feet long, 12 feet wide, and 4 inches thick. One the volume of the cylinder more: multiplying the contractor bid $2225 for the project. A second height by x or multiplying the radius by x. Explain. contractor bid $500 per cubic yard for the new patio SOLUTION: and $700 for removal of the old patio. Which is the a. The oblique cylinder should look like the right less expensive option? Explain. cylinder (same height and size), except that it is SOLUTION: pushed a little to the side, like a slinky. Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume. b. Find the volume of each.

The total cost for the second contractor is about .

The volume of the square prism is greater. c. Therefore, the second contractor is a less expensive Do each scenario. option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders . a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters. Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater. b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume 2 For example, if x = 0.5, then x = 0.25, which is less greater than, less than, or equal to the volume of than x. the cylinder? Explain. ANSWER: c. ANALYTICAL Describe which change affects a. the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is

pushed a little to the side, like a slinky.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater. b. Find the volume of each. c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

The volume of the square prism is greater. c. Do each scenario.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less ANSWER: than x. Francisco; Valerie incorrectly used as the ANSWER: length of one side of the triangular base. Francisco

a. used a different approach, but his solution is correct. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the SOLUTION: radius by x makes the volume x 2 times greater, assuming x > 1. 3 The volume of the can is 20π in . It takes three full 42. CCSS CRITIQUE Franciso and Valerie each cans to fill the container, so the volume of the 3 calculated the volume of an equilateral triangular container is 60π in . prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your a. Choose some basic values for 2 of the sides, and reasoning. then determine the third side. Base: 3 by 5.

ANSWER:

a. rectangular prism

b. square prism

c. triangular prism with a right triangle as the base 3 by 4 by 10π ANSWER: Sample answers:

a. 3 by 5 by 4π

SOLUTION: b. 5 by 5 by

3 The volume of the can is 20π in . It takes three full c. base with legs measuring 3 in. and 4 in., height cans to fill the container, so the volume of the 10π in. container is 60π in 3. 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet a. Choose some basic values for 2 of the sides, and gardening forum. then determine the third side. Base: 3 by 5.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters.

ANSWER: 3 by 4 by 10π Sample answer:

ANSWER: Sample answers:

a. π 3 by 5 by 4

the number of cubic yards of soil needed. The radii must also be equal.

45. OPEN ENDED Draw and label a prism that has a 47. WRITING IN MATH How are the formulas for volume of 50 cubic centimeters. the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest Both formulas involve multiplying the area of the values to choose. base by the height. The base of a prism is a polygon, so the expression representing the area varies, Sample answer: depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area ANSWER: varies, depending on the type of polygon it is. The 2 Sample answer: base of a cylinder is a circle, so its area is πr . 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm 46. REASONING Determine whether the following C 17 cm statement is true or false. Explain. D Two cylinders with the same height and the same 11.5 cm lateral area must have the same volume. SOLUTION: SOLUTION:

True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: B

The radii must also be equal. 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of ANSWER: the tank is 1,122,360 ft3, what is the tank’s radius? True; if two cylinders have the same height and the F 89.4 ft same lateral area, the circular bases must have the G 2 178.8 ft same area. Therefore, πr h is the same for each H 280.9 ft cylinder. J 561.8 ft 47. WRITING IN MATH How are the formulas for SOLUTION: the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr . ANSWER: F 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs 50. SHORT RESPONSE What is the ratio of the area measuring 8 centimeters and 15 centimeters. What is of the circle to the area of the square? the height of the prism? A 34.5 cm B 23 cm C 17 cm D 11.5 cm SOLUTION: SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can

be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5%

property tax. What would be the additional tax ANSWER: amount for a landowner whose property has a taxable value of $85,000? B A $4.25 49. A cylindrical tank used for oil storage has a height B $170 that is half the length of its radius. If the volume of C $425 the tank is 1,122,360 ft3, what is the tank’s radius? D $4250 F 89.4 ft E $42,500 G 178.8 ft SOLUTION: H 280.9 ft Find the 0.5% of $85,000. J 561.8 ft SOLUTION: Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 D $4250 E $42,500 Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem SOLUTION: to find the height h of the triangle. Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , The perimeter is P = 3 × 10 or 30 feet.

where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

Find the perimeter and area of the equilateral

triangle for the base. Use the Pythagorean Theorem Therefore, the surface area of the pyramid is about to find the height h of the triangle. 2 255.4 ft . ANSWER: 212.1 ft2; 255.4 ft2

53.

SOLUTION:

2 So, the lateral area of the pyramid is 126 cm .

So, the area of the base B is ft2.

The surface area S of a regular Find the lateral area L and surface area S of the

regular pyramid. pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is about 212.1 ft2. Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 2 255.4 ft . 54.

ANSWER: SOLUTION: 2 2 The pyramid has a slant height of 15 inches and the 212.1 ft ; 255.4 ft base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

53. SOLUTION:

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER:

2 2 126 cm ; 175 cm Therefore, the surface area of the pyramid is about 758.9 in2.

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

Use a trigonometric ratio to find the measure of the ANSWER: apothem a. 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter.

SOLUTION: Find the lateral area and surface area of the pyramid.

2 So, the lateral area of the pyramid is 472.5 in . The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION: Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

SOLUTION: radius. The area that needs to be coated is the sum of the SOLUTION: lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. ANSWER: SOLUTION: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60.

SOLUTION: The diameter of the circle is about 11.4 m. The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2. ANSWER:

11.4 cm d1 = 12 in. and d2 = 7 + 13 = 20 in. 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 2 120 in

ANSWER: 2 378 m ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: Find the volume of each prism. 108 cm3

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 2.

SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER:

3 108 cm

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. SOLUTION:

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right ANSWER: 26.95 m3

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters

SOLUTION: SOLUTION: If two solids have the same height h and the same If two solids have the same height h and the same cross-sectional area B at every level, then they have cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional an oblique one of the same height and cross sectional area are same. area are same.

ANSWER: 3 ANSWER: 218.4 cm 26.95 m3 Find the volume of each cylinder. Round to the nearest tenth. 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same 5. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and SOLUTION: an oblique one of the same height and cross sectional area are same.

ANSWER: ANSWER: 206.4 ft3 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

6. SOLUTION: 5. If two solids have the same height h and the same cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same.

ANSWER:

206.4 ft3 ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters 6. SOLUTION: SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION: ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters

SOLUTION: ANSWER: 3 410.1 in

9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the ANSWER: lap pool? 1025.4 cm3 A 4000 B 6400 8. a cylinder with a radius of 4.2 inches and a height of C 30,000 7.4 inches D 48,000 SOLUTION: SOLUTION:

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) ANSWER: = 48,000. 3 410.1 in Therefore, the correct choice is D.

9. MULTIPLE CHOICE A rectangular lap pool ANSWER: measures 80 feet long by 20 feet wide. If it needs to D be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the CCSS SENSE-MAKING Find the volume of

lap pool? each prism. A 4000 B 6400 C 30,000 D 48,000 10. SOLUTION: SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. ANSWER: 3 ANSWER: 30 in D CCSS SENSE-MAKING Find the volume of each prism.

11. 10. SOLUTION: SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the The base is a rectangle of length 3 in. and width 2 in. prism is 14 m. The height of the prism is 5 in.

ANSWER: ANSWER: 3 30 in3 539 m

12. 11. SOLUTION: SOLUTION: The base is a right triangle with a leg length of 9 cm The base is a triangle with a base length of 11 m and and the hypotenuse of length 15 cm. the corresponding height of 7 m. The height of the prism is 14 m. Use the Pythagorean Theorem to find the height of the base.

ANSWER: 3 539 m

The height of the prism is 6 cm.

12. SOLUTION: The base is a right triangle with a leg length of 9 cm ANSWER: and the hypotenuse of length 15 cm. 3 324 cm Use the Pythagorean Theorem to find the height of the base.

13. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and The height of the prism is 6 cm. an oblique one of the same height and cross sectional

area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 ANSWER: B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

324 cm3 ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square 13. centimeters SOLUTION: SOLUTION: If two solids have the same height h and the same If two solids have the same height h and the same cross-sectional area B at every level, then they have cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional an oblique one of the same height and cross sectional area are same. area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is ANSWER: 3 ANSWER: 2040 cm 3 58.14 ft 15. a square prism with a base edge of 9.5 inches and a height of 17 inches 14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square SOLUTION: centimeters If two solids have the same height h and the same cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right prism and If two solids have the same height h and the same an oblique one of the same height and cross sectional cross-sectional area B at every level, then they have area are same. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER:

1534.25 in3 ANSWER: 3 CCSS SENSE-MAKING Find the volume of 2040 cm each cylinder. Round to the nearest tenth. 15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: 16. If two solids have the same height h and the same cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right prism and r = 5 yd and h = 18 yd an oblique one of the same height and cross sectional area are same.

ANSWER: 3 ANSWER: 1413.7 yd 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. 17. SOLUTION: r = 6 cm and h = 3.6 cm. 16. SOLUTION: r = 5 yd and h = 18 yd

ANSWER: 407.2 cm3 ANSWER: 3 1413.7 yd

18. SOLUTION: 17. r = 5.5 in. SOLUTION: r = 6 cm and h = 3.6 cm. Use the Pythagorean Theorem to find the height of the cylinder.

ANSWER: 407.2 cm3

Now you can find the volume.

18. SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder. ANSWER: 3 823.0 in

19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and Now you can find the volume. an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm.

ANSWER: 3 823.0 in

ANSWER: 2686.1 mm3

19. 20. PLANTER A planter is in the shape of a SOLUTION: rectangular prism 18 inches long, inches deep, If two solids have the same height h and the same and 12 inches high. What is the volume of potting soil cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and in the planter if the planter is filled to inches an oblique one of the same height and cross sectional below the top? area are same. SOLUTION: r = 7.5 mm and h = 15.2 mm. The planter is to be filled inches below the top, so

ANSWER: 3 2686.1 mm

20. PLANTER A planter is in the shape of a ANSWER: 3 rectangular prism 18 inches long, inches deep, 2740.5 in

and 12 inches high. What is the volume of potting soil 21. SHIPPING A box 18 centimeters by 9 centimeters in the planter if the planter is filled to inches by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 below the top? centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space SOLUTION: in the box? The planter is to be filled inches below the top, so

SOLUTION: The volume of the empty space is the difference of

volumes of the rectangular prism and the cylinders.

ANSWER: 3 2740.5 in

22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION: The volume of the empty space is the difference of SOLUTION: volumes of the rectangular prism and the cylinders. 3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3 521.5 cm Therefore, the diameter of the cylinders should be about 2.52 ft. 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, ANSWER: and 10 cubic feet of sand. To transport the correct 2.52 ft amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. Find the volume of the solid formed by each What should the diameter of the cylinders be? net. SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

23. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism. Therefore, the diameter of the cylinders should be about 2.52 ft. One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height ANSWER: of the base. 2.52 ft Find the volume of the solid formed by each net. The height of the prism is 20 cm.

The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism.

23.

SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the ANSWER: pieces on the ends are the side faces. This is a 3934.9 cm3 triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm.

The volume V of a prism is V = Bh, where B is the 24. area of the base, h is the height of the prism. SOLUTION:

The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique.

ANSWER: The radius is 1.8 m, the height is 4.8 m, and the slant 3934.9 cm3 height is 6 m.

24.

SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. ANSWER: However, since the middle piece is a parallelogram, it 3 48.9 m is oblique.

25. FOOD A cylindrical can of baked potato chips has a The radius is 1.8 m, the height is 4.8 m, and the slant height of 27 centimeters and a radius of 4 height is 6 m. centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the If two solids have the same height h and the same same radius, what is the height of the larger can? cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

ANSWER: 3 48.9 m

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

SOLUTION: 26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. The volume of the smaller can is Describe how each change affects the volume of the

cylinder. The volume of the new can is 130% of the smaller a. The height is tripled.

can, with the same radius. b. The radius is tripled.

c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

The height of the new can will be 35.1 cm. a. When the height is tripled, h = 3h. ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. When the height is tripled, the volume is multiplied by c. Both the radius and the height are tripled. 3. d. The dimensions are exchanged. b. When the radius is tripled, r = 3r.

SOLUTION:

a. When the height is tripled, h = 3h. So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h. Compare to the original volume.

The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. When the height and the radius are tripled, the b. The volume is multiplied by 32 or 9. volume is multiplied by 27. c. 3 The volume is multiplied by 3 or 27.

d. When the dimensions are exchanged, r = 8 and h d. The volume is multiplied by = 5 cm. 27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume.

Compare to the original volume. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a

The volume is multiplied by . plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its ANSWER: weight in pounds? a. The volume is multiplied by 3. SOLUTION: b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density a. First calculate the volume of soil in the pot. Then of the soil sample is the ratio of its weight to its divide the weight of the soil by the volume. volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming

that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound The weight of the soil is the weight of the pot with per square inch is desirable for root growth? Explain. soil minus the weight of the pot. c. If a bag of this soil holds 2.5 cubic feet, what is its W = 20 – 5 = 15 lbs. weight in pounds? SOLUTION: The soil density is thus:

3 3 b. 0.0018 lb/in is close to 0.0019 lb/in so the plant should grow fairly well. a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. c.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. ANSWER: 3 W = 20 – 5 = 15 lbs. a. 0.0019 lb / in

The soil density is thus: b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. Find the volume of each composite solid. Round to the nearest tenth if necessary. c.

28. SOLUTION: The solid is a combination of two rectangular prisms. ANSWER: The base of one rectangular prism is 5 cm by 3 cm 3 a. 0.0019 lb / in and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm. b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round ANSWER: to the nearest tenth if necessary. 225 cm3

28. SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism 29. is 4 cm by 3 cm and the height is 5 cm. SOLUTION:

The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 225 cm3

ANSWER: 3 120 m

29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid 30. is the sum of the volumes of the two rectangular prisms. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

ANSWER: 3 120 m ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the 30. thickness of the base of the holder. The thickness of SOLUTION: the rim of the holder is 1 centimeter. What is the The solid is a combination of a rectangular prism and volume of the rubberized material that makes up the two half cylinders. holder?

SOLUTION: The volume of the rubberized material is the ANSWER: difference between the volumes of the container and the space used for the can. The container has a 260.5 in3 radius of and a height of 11.5 cm. 31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 The empty space used to keep the can has a radius centimeters tall, including 1 centimeter for the of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 thickness of the base of the holder. The thickness of volume V of a cylinder is V = πr h, where r is the the rim of the holder is 1 centimeter. What is the radius of the base and h is the height of the cylinder. volume of the rubberized material that makes up the holder?

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm SOLUTION: The volume of the rubberized material is the Find each measure to the nearest tenth. difference between the volumes of the container and 32. A cylindrical can has a volume of 363 cubic the space used for the can. The container has a centimeters. The diameter of the can is 9 centimeters. What is the height? radius of and a height of 11.5 cm. SOLUTION: The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: ANSWER: 12-4 Volumes of Prisms and Cylinders 3 5.7 cm 304.1 cm

Find each measure to the nearest tenth. 33. A cylinder has a surface area of 144π square inches 32. A cylindrical can has a volume of 363 cubic and a height of 6 inches. What is the volume? centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION: Use the surface area formula to solve for r. SOLUTION:

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches The radius is 6. Find the volume. and a height of 6 inches. What is the volume?

SOLUTION: Use the surface area formula to solve for r.

ANSWER: 678.6 in3

Find the volume. eSolutionsANSWER: Manual - Powered by Cognero Page 11 678.6 in3

34. A rectangular prism has a surface area of 432

square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: ANSWER: 3 Use the surface area formula to find the length of the 576 in base of the prism. 35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter.

Find the volume. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 3 576 in

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. ANSWER: What is the difference in volume between the largest 3 and smallest possible column? Round to the nearest 3,190,680.0 cm tenth cubic centimeter. 36. CCSS MODELING The base of a rectangular SOLUTION: swimming pool is sloped so one end of the pool is 6 The volume will be the highest when the diameter is feet deep and the other end is 3 feet deep, as shown 95 cm and will be the lowest when it is 30 cm.That is in the figure. If the width is 15 feet, find the volume when the radii are 47.5 cm and 15 cm respectively. of water it takes to fill the pool.

Find the difference between the volumes.

ANSWER: 3 1575 ft SOLUTION: The swimming pool is a combination of a rectangular 37. CHANGING DIMENSIONS A soy milk company prism and a trapezoidal prism. The base of the is planning a promotion in which the volume of soy rectangular prism is 6 ft by 10 ft and the height is 15 milk in each container will be increased by 25%. The ft. The bases of the trapezoidal prism are 6 ft and 3 company wants the base of the container to stay the ft long and the height of the base is 10 ft. The height same. What will be the height of the new containers? of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms.

SOLUTION: Find the volume of the original container.

ANSWER: 3 1575 ft

The volume of the new container is 125% of the CHANGING DIMENSIONS 37. A soy milk company original container, with the same base dimensions. is planning a promotion in which the volume of soy Use 1.25V and B to find h. milk in each container will be increased by 25%. The company wants the base of the container to stay the same. What will be the height of the new containers?

ANSWER:

SOLUTION: Find the volume of the original container. 38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3)

The volume of the new container is 125% of the SOLUTION: original container, with the same base dimensions. Sample answers: Use 1.25V and B to find h. For any cylindrical container, we have the following equation for volume:

ANSWER: The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable 38. DESIGN Sketch and label (in inches) three different radius, say 1.85 in, and solve for the height. designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume: If we choose a height of say 4 in., then we can solve for the radius.

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = If we choose a height of say 4 in., then we can solve 2.5 in. for the radius.

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

ANSWER: Sample answers:

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe ANSWER: 3 the base area and height of each triangular prism. 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second SOLUTION: contractor bid $500 per cubic yard for the new patio The base of the prism can be divided into 5 and $700 for removal of the old patio. Which is the congruent triangles of a base 8 cm and the less expensive option? Explain. corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height SOLUTION: 10 cm. Find the base area of each triangular prism. Convert all of the dimensions to yards.

20 feet = yd Therefore, the volume of the pentagonal prism is 12 feet = 4 yd

4 in. = yd ANSWER: 3 Find the volume. 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

and $700 for removal of the old patio. Which is the less expensive option? Explain. The total cost for the second contractor is about SOLUTION: . Convert all of the dimensions to yards. Therefore, the second contractor is a less expensive 20 feet = yd option. 12 feet = 4 yd ANSWER: 4 in. = yd Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50. Find the volume. 41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders . a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume The total cost for the second contractor is about greater than, less than, or equal to the volume of . the cylinder? Explain.

Therefore, the second contractor is a less expensive c. ANALYTICAL Describe which change affects option. the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. ANSWER: SOLUTION: Because 2.96 yd3 of concrete are needed, the a. The oblique cylinder should look like the right second contractor is less expensive at $2181.50. cylinder (same height and size), except that it is pushed a little to the side, like a slinky. 41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders . a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. Find the volume of each. b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky. The volume of the square prism is greater. c. Do each scenario.

b. Find the volume of each.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less than x.

The volume of the square prism is greater. ANSWER: c. Do each scenario. a.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is Assuming x > 1, multiplying the radius by x makes greater. 2 the volume x times greater. c. Multiplying the radius by x ; since the volume is 2 For example, if x = 0.5, then x = 0.25, which is less represented by π r 2 h, multiplying the height by x than x. makes the volume x times greater. Multiplying the ANSWER: radius by x makes the volume x 2 times greater, a. assuming x > 1. 42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights SOLUTION: are the same, the volume of the square prism is Francisco; Valerie incorrectly used as the greater. length of one side of the triangular base. Francisco used a different approach, but his solution is correct. c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x Francisco used the standard formula for the volume makes the volume x times greater. Multiplying the of a solid, V = Bh. The area of the base, B, is one- radius by x makes the volume x 2 times greater, half the apothem multiplied by the perimeter of the assuming x > 1. base.

42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning. ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

43. CHALLENGE A cylindrical can is used to fill a SOLUTION: container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the Francisco; Valerie incorrectly used as the container if it is each of the following shapes. length of one side of the triangular base. Francisco

used a different approach, but his solution is correct. a. rectangular prism Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- b. square prism half the apothem multiplied by the perimeter of the base. c. triangular prism with a right triangle as the base

ANSWER: Francisco; Valerie incorrectly used as the SOLUTION: length of one side of the triangular base. Francisco used a different approach, but his solution is correct. 3 The volume of the can is 20π in . It takes three full 43. CHALLENGE A cylindrical can is used to fill a cans to fill the container, so the volume of the 3 container with liquid. It takes three full cans to fill the container is 60π in . container. Describe possible dimensions of the container if it is each of the following shapes. a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. a. rectangular prism

b. square prism

c. triangular prism with a right triangle as the base

3 by 5 by 4π

b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. SOLUTION:

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the container is 60π in 3.

a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5.

5 by 5 by

c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4.

3 by 5 by 4π

b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5.

3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. 10π in. 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? 3 by 4 by 10π SOLUTION: ANSWER: Sample answer: The nursery means a cubic yard, Sample answers: 3 which is 3 or 27 cubic feet. Find the volume of your

garden in cubic feet and divide by 27 to determine a. 3 by 5 by 4π the number of cubic yards of soil needed.

ANSWER: b. 5 by 5 by Sample answer: The nursery means a cubic yard,

which is 33 or 27 cubic feet. Find the volume of your c. base with legs measuring 3 in. and 4 in., height garden in cubic feet and divide by 27 to determine 10π in. the number of cubic yards of soil needed. WRITING IN MATH 44. Write a helpful response to 45. OPEN ENDED Draw and label a prism that has a the following question posted on an Internet volume of 50 cubic centimeters. gardening forum. SOLUTION: I am new to gardening. The nursery will deliver a Choose 3 values that multiply to make 50. The truckload of soil, which they say is 4 yards. I factors of 50 are 2, 5, 5, so these are the simplest know that a yard is 3 feet, but what is a yard of values to choose. soil? How do I know what to order?

SOLUTION: Sample answer: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: ANSWER: Sample answer: The nursery means a cubic yard, Sample answer: which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. SOLUTION: 46. REASONING Determine whether the following Choose 3 values that multiply to make 50. The statement is true or false. Explain. factors of 50 are 2, 5, 5, so these are the simplest Two cylinders with the same height and the same values to choose. lateral area must have the same volume.

Sample answer: SOLUTION: True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: Sample answer:

The radii must also be equal.

ANSWER: 46. REASONING Determine whether the following statement is true or false. Explain. True; if two cylinders have the same height and the Two cylinders with the same height and the same same lateral area, the circular bases must have the 2 lateral area must have the same volume. same area. Therefore, πr h is the same for each cylinder. SOLUTION:

True; if two cylinders have the same height (h1 = h2) 47. WRITING IN MATH How are the formulas for and the same lateral area (L = L ), the circular the volume of a prism and the volume of a cylinder 1 2 similar? How are they different? bases must have the same area. SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the The radii must also be equal. area of the base by the height. The base of a prism is ANSWER: a polygon, so the expression representing the area varies, depending on the type of polygon it is. The True; if two cylinders have the same height and the 2 same lateral area, the circular bases must have the base of a cylinder is a circle, so its area is πr . 2 same area. Therefore, πr h is the same for each 48. The volume of a triangular prism is 1380 cubic cylinder. centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is 47. WRITING IN MATH How are the formulas for the height of the prism? the volume of a prism and the volume of a cylinder A 34.5 cm similar? How are they different? B 23 cm SOLUTION: C 17 cm Both formulas involve multiplying the area of the D 11.5 cm base by the height. The base of a prism is a polygon, so the expression representing the area varies, SOLUTION: depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr . 48. The volume of a triangular prism is 1380 cubic ANSWER: centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is B the height of the prism? 49. A cylindrical tank used for oil storage has a height A 34.5 cm that is half the length of its radius. If the volume of B 23 cm the tank is 1,122,360 ft3, what is the tank’s radius? C 17 cm F 89.4 ft D 11.5 cm G 178.8 ft SOLUTION: H 280.9 ft J 561.8 ft SOLUTION:

ANSWER: B 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of

the tank is 1,122,360 ft3, what is the tank’s radius? F 89.4 ft ANSWER: G 178.8 ft F H 280.9 ft J 561.8 ft 50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square? SOLUTION:

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

ANSWER: 51. SAT/ACT A county proposes to enact a new 0.5% F property tax. What would be the additional tax amount for a landowner whose property has a 50. SHORT RESPONSE What is the ratio of the area taxable value of $85,000? of the circle to the area of the square? A $4.25 B $170 C $425 D $4250

E $42,500 SOLUTION: SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can Find the 0.5% of $85,000. be written as shown.

Therefore, the correct choice is C.

ANSWER: ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if 51. SAT/ACT A county proposes to enact a new 0.5% necessary. property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 52. C $425 D $4250 SOLUTION: E $42,500 The lateral area L of a regular pyramid is , SOLUTION: where is the slant height and P is the perimeter of Find the 0.5% of $85,000. the base.

The slant height is the height of each of the Therefore, the correct choice is C. congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem The perimeter is P = 3 × 10 or 30 feet. to find the height h of the triangle.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the

regular pyramid.

The perimeter is P = 3 × 10 or 30 feet. So, the lateral area of the pyramid is about 212.1 ft2.

So, the area of the base B is ft2. Therefore, the surface area of the pyramid is about 2 Find the lateral area L and surface area S of the 255.4 ft . regular pyramid. ANSWER: 212.1 ft2; 255.4 ft2

So, the lateral area of the pyramid is about 212.1 ft2. 53. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant

height is 9 cm. Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 212.1 ft2; 255.4 ft2

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular pyramid is , where 53. L is the lateral area and B is the area of the base.

SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant

height is 9 cm. Therefore, the surface area of the pyramid is 175

cm2.

ANSWER: 2 2 126 cm ; 175 cm

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular 54. pyramid is , where L is the lateral area and B is the area of the base. SOLUTION: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

Use a trigonometric ratio to find the measure of the apothem a.

54. SOLUTION: The pyramid has a slant height of 15 inches and the

base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°. Find the lateral area and surface area of the pyramid.

So, the lateral area of the pyramid is 472.5 in2.

Use a trigonometric ratio to find the measure of the apothem a.

Therefore, the surface area of the pyramid is about 758.9 in2. Find the lateral area and surface area of the pyramid. ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of So, the lateral area of the pyramid is 472.5 in2. the inside of the pan that needs to be coated?

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 Therefore, the surface area of the pyramid is about 9)(2) + 13(9) = 205 in . 758.9 in2. ANSWER: 205 in2 ANSWER: 472.5 in2; 758.9 in2 Find the indicated measure. Round to the nearest tenth. 55. BAKING Many baking pans are given a special 56. The area of a circle is 54 square meters. Find the nonstick coating. A rectangular cake pan is 9 inches diameter. by 13 inches by 2 inches deep. What is the area of SOLUTION: the inside of the pan that needs to be coated?

SOLUTION: The area that needs to be coated is the sum of the

lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in . The diameter of the circle is about 8.3 m.

ANSWER: ANSWER: 205 in2 8.3 m

Find the indicated measure. Round to the 57. Find the diameter of a circle with an area of 102 nearest tenth. square centimeters. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION: SOLUTION:

The diameter of the circle is about 11.4 m.

The diameter of the circle is about 8.3 m. ANSWER: 11.4 cm ANSWER: 8.3 m 58. The area of a circle is 191 square feet. Find the radius. 57. Find the diameter of a circle with an area of 102 SOLUTION: square centimeters. SOLUTION:

ANSWER: The diameter of the circle is about 11.4 m. 7.8 ft

ANSWER: 59. Find the radius of a circle with an area of 271 square

11.4 cm inches. SOLUTION: 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 9.3 in. ANSWER: 7.8 ft Find the area of each trapezoid, rhombus, or kite. 59. Find the radius of a circle with an area of 271 square inches. SOLUTION:

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 9.3 in.

Find the area of each trapezoid, rhombus, or kite. ANSWER: 2 120 in

60. SOLUTION: The area A of a kite is one half the product of the 61. lengths of its diagonals, d and d . 1 2 SOLUTION:

The area A of a trapezoid is one half the product of d1 = 12 in. and d2 = 7 + 13 = 20 in. the height h and the sum of the lengths of its bases,

b1 and b2.

ANSWER: 2 120 in ANSWER: 2 378 m

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

Find the volume of each prism.

2. SOLUTION: 1. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm . ANSWER: 3 396 in ANSWER: 108 cm3 3. the oblique rectangular prism shown at the right

SOLUTION: If two solids have the same height h and the same 2. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and SOLUTION: an oblique one of the same height and cross sectional The volume V of a prism is V = Bh, where B is the area are same. area of a base and h is the height of the prism.

ANSWER: 3 26.95 m

ANSWER: 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters 3 396 in SOLUTION: 3. the oblique rectangular prism shown at the right If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and

an oblique one of the same height and cross sectional area are same. ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 5. 26.95 m3 SOLUTION: 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 3 206.4 ft

ANSWER: 6. 3 218.4 cm SOLUTION: If two solids have the same height h and the same Find the volume of each cylinder. Round to the cross-sectional area B at every level, then they have nearest tenth. the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same.

5. SOLUTION:

ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a ANSWER: height of 5.1 centimeters 206.4 ft3 SOLUTION:

6. SOLUTION: ANSWER: If two solids have the same height h and the same 1025.4 cm3 cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder 8. a cylinder with a radius of 4.2 inches and a height of and an oblique one of the same height and cross 7.4 inches

sectional area are same. SOLUTION:

ANSWER: ANSWER: 3 3 410.1 in 1357.2 m MULTIPLE CHOICE 7. a cylinder with a diameter of 16 centimeters and a 9. A rectangular lap pool height of 5.1 centimeters measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds SOLUTION: 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000 D 48,000 SOLUTION: ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches Each cubic foot holds 7.5 gallons of water. So, the SOLUTION: amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: D ANSWER: CCSS SENSE-MAKING Find the volume of 3 410.1 in each prism.

9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 10. 7.5 gallons, how many gallons will it take to fill the SOLUTION: lap pool? The base is a rectangle of length 3 in. and width 2 in. A 4000 The height of the prism is 5 in. B 6400 C 30,000 D 48,000 SOLUTION:

ANSWER: 3 30 in Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: 11. D SOLUTION: CCSS SENSE-MAKING Find the volume of The base is a triangle with a base length of 11 m and each prism. the corresponding height of 7 m. The height of the prism is 14 m.

10. SOLUTION:

The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. ANSWER: 3 539 m

ANSWER: 12. 30 in3 SOLUTION: The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm.

Use the Pythagorean Theorem to find the height of the base. 11. SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the prism is 14 m.

The height of the prism is 6 cm.

ANSWER: 3 539 m

ANSWER: 324 cm3

12. SOLUTION: The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. 13. Use the Pythagorean Theorem to find the height of the base. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

The height of the prism is 6 cm.

ANSWER: 58.14 ft3

ANSWER: 14. an oblique hexagonal prism with a height of 15 3 centimeters and with a base area of 136 square 324 cm centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 13. area are same.

3 The volume V of a prism is V = Bh, where B is the 2040 cm area of a base and h is the height of the prism. 15. a square prism with a base edge of 9.5 inches and a

2 height of 17 inches B = 11.4 ft and h = 5.1 ft. Therefore, the volume is SOLUTION: If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 3 the same volume. So, the volume of a right prism and 58.14 ft an oblique one of the same height and cross sectional area are same. 14. an oblique hexagonal prism with a height of 15

centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 3 area are same. 1534.25 in CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 16. 3 2040 cm SOLUTION: r = 5 yd and h = 18 yd 15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: 3 area are same. 1413.7 yd

17. ANSWER: SOLUTION: 1534.25 in3 r = 6 cm and h = 3.6 cm.

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. ANSWER: SOLUTION: 407.2 cm3 r = 5 yd and h = 18 yd

18. ANSWER: SOLUTION: 3 r = 5.5 in. 1413.7 yd Use the Pythagorean Theorem to find the height of the cylinder.

17. SOLUTION: r = 6 cm and h = 3.6 cm.

Now you can find the volume. ANSWER: 407.2 cm3

18. ANSWER: 3 SOLUTION: 823.0 in r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

r = 7.5 mm and h = 15.2 mm.

Now you can find the volume.

ANSWER:

2686.1 mm3

ANSWER: 20. PLANTER A planter is in the shape of a 3 823.0 in rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches

19. below the top? SOLUTION: SOLUTION: If two solids have the same height h and the same The planter is to be filled inches below the top, so cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm.

ANSWER: 3 2740.5 in

ANSWER: 21. SHIPPING A box 18 centimeters by 9 centimeters 2686.1 mm3 by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 20. PLANTER A planter is in the shape of a centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space rectangular prism 18 inches long, inches deep, in the box? and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top?

SOLUTION: The planter is to be filled inches below the top, so SOLUTION: The volume of the empty space is the difference of

volumes of the rectangular prism and the cylinders.

ANSWER: 3 ANSWER: 2740.5 in 521.5 cm3 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two 22. SANDCASTLES In a sandcastle competition, cylindrical candles. Each candle has a diameter of 9 contestants are allowed to use only water, shovels, centimeters and a height of 15 centimeters, as shown and 10 cubic feet of sand. To transport the correct at the right. What is the volume of the empty space amount of sand, they want to create cylinders that in the box? are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER:

2.52 ft ANSWER: 3 Find the volume of the solid formed by each 521.5 cm net. 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? 23. SOLUTION: SOLUTION:

3 The middle piece of the net is the front of the solid. V = 10 ft and h = 2 ft Use the formula to find r. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

Therefore, the diameter of the cylinders should be The height of the prism is 20 cm. about 2.52 ft.

ANSWER: The volume V of a prism is V = Bh, where B is the 2.52 ft area of the base, h is the height of the prism.

Find the volume of the solid formed by each net.

ANSWER: 3934.9 cm3 23. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height 24. of the base. SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece The height of the prism is 20 cm. were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it The volume V of a prism is V = Bh, where B is the is oblique. area of the base, h is the height of the prism. The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

ANSWER: 3 24. 48.9 m SOLUTION: 25. FOOD A cylindrical can of baked potato chips has a The circular bases at the top and bottom of the net height of 27 centimeters and a radius of 4 indicate that this is a cylinder. If the middle piece centimeters. A new can is advertised as being 30% were a rectangle, then the prism would be right. larger than the regular can. If both cans have the However, since the middle piece is a parallelogram, it same radius, what is the height of the larger can? is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

If two solids have the same height h and the same cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right prism and The volume of the smaller can is an oblique one of the same height and cross sectional area are same. The volume of the new can is 130% of the smaller can, with the same radius.

ANSWER: 3 48.9 m The height of the new can will be 35.1 cm. 25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 ANSWER: centimeters. A new can is advertised as being 30% 35.1 cm larger than the regular can. If both cans have the same radius, what is the height of the larger can? 26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. SOLUTION: d. The dimensions are exchanged. The volume of the smaller can is SOLUTION: The volume of the new can is 130% of the smaller can, with the same radius.

a. When the height is tripled, h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm When the height is tripled, the volume is multiplied by 3. 26. CHANGING DIMENSIONS A cylinder has a b. When the radius is tripled, r = 3r. radius of 5 centimeters and a height of 8 centimeters.

Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION: So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

a. When the height is tripled, h = 3h.

When the height and the radius are tripled, the volume is multiplied by 27. When the height is tripled, the volume is multiplied by 3. d. When the dimensions are exchanged, r = 8 and h b. When the radius is tripled, r = 3r. = 5 cm.

Compare to the original volume.

So, when the radius is tripled, the volume is multiplied by 9.

The volume is multiplied by . c. When the height and the radius are tripled, r = 3r and h = 3h. ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk When the height and the radius are tripled, the density of a potting soil to assess how well a specific volume is multiplied by 27. plant will grow in it. The density of the soil sample is the ratio of its weight to its d. When the dimensions are exchanged, r = 8 and h volume. = 5 cm. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its Compare to the original volume. weight in pounds?

SOLUTION:

The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. 2 b. The volume is multiplied by 3 or 9. a. 3 First calculate the volume of soil in the pot. Then c. The volume is multiplied by 3 or 27. divide the weight of the soil by the volume.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume. The weight of the soil is the weight of the pot with soil minus the weight of the pot. a. If the weight of the container with the soil is 20 W = 20 – 5 = 15 lbs. pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming The soil density is thus: that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain.

c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant SOLUTION: should grow fairly well.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the 3 bulk density of 0.0019 lb / in is close to the desired 3 The weight of the soil is the weight of the pot with bulk density of 0.0018 lb / in . soil minus the weight of the pot. W = 20 – 5 = 15 lbs. c. 8.3 lb

The soil density is thus: Find the volume of each composite solid. Round to the nearest tenth if necessary.

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. 28. c. SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the ANSWER: 3 bulk density of 0.0019 lb / in is close to the desired 3 3 225 cm bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

29. SOLUTION: 28. The solid is a combination of a rectangular prism and SOLUTION: a right triangular prism. The total volume of the solid The solid is a combination of two rectangular prisms. is the sum of the volumes of the two rectangular The base of one rectangular prism is 5 cm by 3 cm prisms. and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: 3 ANSWER: 225 cm 3 120 m

30. SOLUTION: 29. The solid is a combination of a rectangular prism and

SOLUTION: two half cylinders.

The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 260.5 in

31. MANUFACTURING A can 12 centimeters tall fits ANSWER: into a rubberized cylindrical holder that is 11.5 3 centimeters tall, including 1 centimeter for the 120 m thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

30. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a

radius of and a height of 11.5 cm.

The empty space used to keep the can has a radius

of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 ANSWER: volume V of a cylinder is V = πr h, where r is the 260.5 in3 radius of the base and h is the height of the cylinder.

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5

centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of Therefore, the volume of the rubberized material is the rim of the holder is 1 centimeter. What is the 3 volume of the rubberized material that makes up the about 304.1 cm . holder? ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height?

SOLUTION: SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

ANSWER: 5.7 cm

Therefore, the volume of the rubberized material is 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? about 304.1 cm3. SOLUTION: ANSWER: Use the surface area formula to solve for r. 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

The radius is 6. Find the volume.

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume?

SOLUTION: Use the surface area formula to solve for r. ANSWER: 678.6 in3

The radius is 6. Find the volume.

Find the volume.

ANSWER: ANSWER: 12-4 Volumes of Prisms and Cylinders 3 678.6 in3 576 in

34. A rectangular prism has a surface area of 432 35. ARCHITECTURE A cylindrical stainless steel square inches, a height of 6 inches, and a width of 12 column is used to hide a ventilation system in a new inches. What is the volume? building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 SOLUTION: centimeters. The height is to be 500 centimeters. Use the surface area formula to find the length of the What is the difference in volume between the largest base of the prism. and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

Find the volume.

ANSWER: 3 ANSWER: 576 in 3,190,680.0 cm3 35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new 36. CCSS MODELING The base of a rectangular building. According to the specifications, the diameter swimming pool is sloped so one end of the pool is 6 of the column can be between 30 centimeters and 95 feet deep and the other end is 3 feet deep, as shown centimeters. The height is to be 500 centimeters. in the figure. If the width is 15 feet, find the volume What is the difference in volume between the largest of water it takes to fill the pool. and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is

95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively. SOLUTION: The swimming pool is a combination of a rectangular Find the difference between the volumes. prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms.

eSolutionsANSWER: Manual - Powered by Cognero Page 12 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown ANSWER: 3 in the figure. If the width is 15 feet, find the volume 1575 ft of water it takes to fill the pool. 37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the same. What will be the height of the new containers? SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two SOLUTION: prisms. Find the volume of the original container.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

ANSWER: 3 1575 ft

For any cylindrical container, we have the following equation for volume:

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable

radius, say 1.85 in, and solve for the height.

ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each 3 If we choose a height of say 4 in., then we can solve drawing. (1 cup ≈ 14.4375 in ) for the radius. SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

The last equation gives us a relation between the For any rectangular container, the volume equation is: radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height. Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

If we choose a height of say 4 in., then we can solve for the radius.

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

ANSWER: Sample answers:

ANSWER: 39. Find the volume of the regular pentagonal prism by Sample answers: dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

39. Find the volume of the regular pentagonal prism by 40. PATIOS Mr. Thomas is planning to remove an old dividing it into five equal triangular prisms. Describe patio and install a new rectangular concrete patio 20 the base area and height of each triangular prism. feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards. SOLUTION: The base of the prism can be divided into 5 20 feet = yd congruent triangles of a base 8 cm and the 12 feet = 4 yd corresponding height 5.5 cm. So, the pentagonal 4 in. = yd prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism. Find the volume.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old The total cost for the second contractor is about patio and install a new rectangular concrete patio 20 . feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second Therefore, the second contractor is a less expensive contractor bid $500 per cubic yard for the new patio option. and $700 for removal of the old patio. Which is the less expensive option? Explain. ANSWER: 3 SOLUTION: Because 2.96 yd of concrete are needed, the Convert all of the dimensions to yards. second contractor is less expensive at $2181.50.

MULTIPLE REPRESENTATIONS 41. In this 20 feet = yd problem, you will investigate right and oblique 12 feet = 4 yd cylinders 4 in. = yd . a. GEOMETRIC Draw a right cylinder and an Find the volume. oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the

height by x or multiplying the radius by x. Explain. The total cost for the second contractor is about SOLUTION: . a. The oblique cylinder should look like the right cylinder (same height and size), except that it is Therefore, the second contractor is a less expensive pushed a little to the side, like a slinky. option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique b. Find the volume of each. cylinders . a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the The volume of the square prism is greater. height by x or multiplying the radius by x. Explain. c. Do each scenario. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Find the volume of each. Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less than x.

ANSWER:

The volume of the square prism is greater. c. Do each scenario.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 2 m has an area of 9π or 28.3 m . Since the heights are the same, the volume of the square prism is greater.

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, Assuming x > 1, multiplying the radius by x makes 2 assuming x > 1. the volume x times greater. 42. CCSS CRITIQUE Franciso and Valerie each For example, if x = 0.5, then x2 = 0.25, which is less calculated the volume of an equilateral triangular than x. prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your ANSWER: reasoning. a.

SOLUTION: Francisco; Valerie incorrectly used as the

length of one side of the triangular base. Francisco used a different approach, but his solution is correct. b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 Francisco used the standard formula for the volume m has an area of 9π or 28.3 m2. Since the heights of a solid, V = Bh. The area of the base, B, is one- are the same, the volume of the square prism is half the apothem multiplied by the perimeter of the greater. base.

c. Multiplying the radius by x ; since the volume is

π represented by r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, ANSWER: assuming x > 1. Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco 42. CCSS CRITIQUE Franciso and Valerie each used a different approach, but his solution is correct. calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 43. CHALLENGE A cylindrical can is used to fill a units. Is either of them correct? Explain your container with liquid. It takes three full cans to fill the

reasoning. container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

SOLUTION: b. square prism

Francisco; Valerie incorrectly used as the c. triangular prism with a right triangle as the base length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base. SOLUTION:

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the 3 ANSWER: container is 60π in .

Francisco; Valerie incorrectly used as the a. Choose some basic values for 2 of the sides, and length of one side of the triangular base. Francisco then determine the third side. Base: 3 by 5. used a different approach, but his solution is correct.

a. rectangular prism 3 by 5 by 4π

b. square prism b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. c. triangular prism with a right triangle as the base

SOLUTION:

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the 5 by 5 by

container is 60π in 3. c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5.

3 by 4 by 10π 3 by 5 by 4π ANSWER: b. Choose some basic values for 2 of the sides, and Sample answers: then determine the third side. Base: 5 by 5. a. 3 by 5 by 4π

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet 5 by 5 by gardening forum.

c. Choose some basic values for 2 of the sides, and I am new to gardening. The nursery will deliver a then determine the third side. Base: Legs: 3 by 4. truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. 3 by 4 by 10π ANSWER: ANSWER: Sample answer: The nursery means a cubic yard, Sample answers: which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine a. 3 by 5 by 4π the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a b. 5 by 5 by volume of 50 cubic centimeters.

SOLUTION: c. base with legs measuring 3 in. and 4 in., height Choose 3 values that multiply to make 50. The 10π in. factors of 50 are 2, 5, 5, so these are the simplest values to choose. 44. WRITING IN MATH Write a helpful response to

the following question posted on an Internet Sample answer: gardening forum.

ANSWER: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your 46. REASONING Determine whether the following garden in cubic feet and divide by 27 to determine statement is true or false. Explain. the number of cubic yards of soil needed. Two cylinders with the same height and the same lateral area must have the same volume. 45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. SOLUTION: True; if two cylinders have the same height (h = h ) SOLUTION: 1 2 Choose 3 values that multiply to make 50. The and the same lateral area (L1 = L2), the circular factors of 50 are 2, 5, 5, so these are the simplest bases must have the same area. values to choose.

Sample answer:

ANSWER: The radii must also be equal. Sample answer: ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 same area. Therefore, πr h is the same for each cylinder.

47. WRITING IN MATH How are the formulas for 46. REASONING Determine whether the following the volume of a prism and the volume of a cylinder statement is true or false. Explain. similar? How are they different? Two cylinders with the same height and the same lateral area must have the same volume. SOLUTION: Both formulas involve multiplying the area of the SOLUTION: base by the height. The base of a prism is a polygon, True; if two cylinders have the same height (h1 = h2) so the expression representing the area varies, and the same lateral area (L = L ), the circular depending on the type of polygon it is. The base of a 1 2 2 bases must have the same area. cylinder is a circle, so its area is πr . ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr .

47. WRITING IN MATH How are the formulas for SOLUTION: the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the ANSWER: area of the base by the height. The base of a prism is B a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 49. A cylindrical tank used for oil storage has a height 2 base of a cylinder is a circle, so its area is πr . that is half the length of its radius. If the volume of the tank is 1,122,360 ft3, what is the tank’s radius? 48. The volume of a triangular prism is 1380 cubic F 89.4 ft centimeters. Its base is a right triangle with legs G 178.8 ft measuring 8 centimeters and 15 centimeters. What is H 280.9 ft the height of the prism? J 561.8 ft A 34.5 cm B 23 cm SOLUTION: C 17 cm D 11.5 cm SOLUTION:

ANSWER: ANSWER: B F 49. A cylindrical tank used for oil storage has a height 50. SHORT RESPONSE What is the ratio of the area that is half the length of its radius. If the volume of of the circle to the area of the square? the tank is 1,122,360 ft3, what is the tank’s radius? F 89.4 ft G 178.8 ft

H 280.9 ft J 561.8 ft SOLUTION: The radius of the circle is 2x and the length of each SOLUTION: side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 ANSWER: C $425 F D $4250 E $42,500 50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square? SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C. SOLUTION: ANSWER: The radius of the circle is 2x and the length of each C side of the square is 4x. So, the ratio of the areas can be written as shown. Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary. ANSWER:

52. 51. SAT/ACT A county proposes to enact a new 0.5% SOLUTION: property tax. What would be the additional tax amount for a landowner whose property has a The lateral area L of a regular pyramid is , taxable value of $85,000? A $4.25 where is the slant height and P is the perimeter of B $170 the base. C $425 D $4250 The slant height is the height of each of the E $42,500 congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of Find the perimeter and area of the equilateral the base. triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

The perimeter is P = 3 × 10 or 30 feet.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

2 So, the lateral area of the pyramid is about 212.1 ft .

The perimeter is P = 3 × 10 or 30 feet.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 212.1 ft2; 255.4 ft2 So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

53. SOLUTION:

The lateral area L of a regular pyramid is ,

So, the lateral area of the pyramid is about 212.1 ft2. where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

Therefore, the surface area of the pyramid is about So, the lateral area of the pyramid is 126 cm2. 2 255.4 ft .

ANSWER: The surface area S of a regular 212.1 ft2; 255.4 ft2 pyramid is , where L is the lateral area and B is the area of the base.

53.

SOLUTION: Therefore, the surface area of the pyramid is 175 2 The lateral area L of a regular pyramid is , cm . where is the slant height and P is the perimeter of ANSWER: 2 2 the base. 126 cm ; 175 cm Here, the base is a square of side 7 cm and the slant height is 9 cm.

54. SOLUTION: So, the lateral area of the pyramid is 126 cm2. The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the The surface area S of a regular angle formed in the triangle below is 30°. pyramid is , where L is the lateral area and B is the area of the base.

Therefore, the surface area of the pyramid is 175 cm2. Use a trigonometric ratio to find the measure of the apothem a. ANSWER: 2 2 126 cm ; 175 cm

54. Find the lateral area and surface area of the pyramid. SOLUTION: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

So, the lateral area of the pyramid is 472.5 in2.

Use a trigonometric ratio to find the measure of the Therefore, the surface area of the pyramid is about apothem a. 2 758.9 in .

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special Find the lateral area and surface area of the nonstick coating. A rectangular cake pan is 9 inches pyramid. by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

SOLUTION: So, the lateral area of the pyramid is 472.5 in2. The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the indicated measure. Round to the Therefore, the surface area of the pyramid is about nearest tenth. 2 56. The area of a circle is 54 square meters. Find the 758.9 in . diameter.

SOLUTION: ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

The diameter of the circle is about 8.3 m.

SOLUTION: ANSWER: The area that needs to be coated is the sum of the 8.3 m lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 57. Find the diameter of a circle with an area of 102 2 9)(2) + 13(9) = 205 in . square centimeters.

ANSWER: SOLUTION: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius.

SOLUTION: The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches. SOLUTION:

The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius.

SOLUTION:

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

ANSWER: 7.8 ft 60. 59. Find the radius of a circle with an area of 271 square SOLUTION: inches. The area A of a kite is one half the product of the lengths of its diagonals, d and d . SOLUTION: 1 2

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 ANSWER: 120 in 9.3 in. Find the area of each trapezoid, rhombus, or kite.

61. SOLUTION: The area A of a trapezoid is one half the product of 60. the height h and the sum of the lengths of its bases,

SOLUTION: b1 and b2. The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 378 m

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the Find the volume of each prism. area of a base and h is the height of the prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right 3 The volume is 108 cm .

ANSWER: 3 108 cm SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 26.95 m3

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and 3 396 in an oblique one of the same height and cross sectional area are same. 3. the oblique rectangular prism shown at the right

SOLUTION: ANSWER: If two solids have the same height h and the same 3 cross-sectional area B at every level, then they have 218.4 cm the same volume. So, the volume of a right prism and Find the volume of each cylinder. Round to the an oblique one of the same height and cross sectional nearest tenth. area are same.

5. SOLUTION:

ANSWER: 26.95 m3

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: ANSWER: If two solids have the same height h and the same 206.4 ft3 cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

Find the volume of each cylinder. Round to the nearest tenth.

5. ANSWER: 3 SOLUTION: 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION:

ANSWER: 206.4 ft3

ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of

6. 7.4 inches SOLUTION: SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 410.1 in

9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to ANSWER: be filled to four feet deep and each cubic foot holds 3 7.5 gallons, how many gallons will it take to fill the 1357.2 m lap pool? A 4000 7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters B 6400 C 30,000 SOLUTION: D 48,000 SOLUTION:

ANSWER: Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) 3 1025.4 cm = 48,000.

8. a cylinder with a radius of 4.2 inches and a height of Therefore, the correct choice is D. 7.4 inches SOLUTION: ANSWER: D CCSS SENSE-MAKING Find the volume of each prism.

ANSWER: 10. 3 410.1 in SOLUTION: The base is a rectangle of length 3 in. and width 2 in. 9. MULTIPLE CHOICE A rectangular lap pool The height of the prism is 5 in. measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000

B 6400 C 30,000 ANSWER: D 48,000 30 in3 SOLUTION:

Each cubic foot holds 7.5 gallons of water. So, the 11. amount of water required to fill the pool is 6400(7.5) = 48,000. SOLUTION: The base is a triangle with a base length of 11 m and Therefore, the correct choice is D. the corresponding height of 7 m. The height of the prism is 14 m. ANSWER:

D CCSS SENSE-MAKING Find the volume of each prism.

ANSWER: 3 10. 539 m SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

12. SOLUTION: The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. ANSWER: 3 30 in Use the Pythagorean Theorem to find the height of the base.

11. SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the

prism is 14 m. The height of the prism is 6 cm.

ANSWER: ANSWER: 3 3 324 cm 539 m

12. 13. SOLUTION: SOLUTION: The base is a right triangle with a leg length of 9 cm If two solids have the same height h and the same and the hypotenuse of length 15 cm. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and Use the Pythagorean Theorem to find the height of an oblique one of the same height and cross sectional the base. area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

ANSWER:

58.14 ft3 The height of the prism is 6 cm. 14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters

13. SOLUTION: ANSWER: If two solids have the same height h and the same 3 cross-sectional area B at every level, then they have 2040 cm the same volume. So, the volume of a right prism and 15. a square prism with a base edge of 9.5 inches and a an oblique one of the same height and cross sectional height of 17 inches area are same. SOLUTION: The volume V of a prism is V = Bh, where B is the If two solids have the same height h and the same area of a base and h is the height of the prism. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is an oblique one of the same height and cross sectional area are same.

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters ANSWER: 3 SOLUTION: 1534.25 in If two solids have the same height h and the same CCSS SENSE-MAKING Find the volume of cross-sectional area B at every level, then they have each cylinder. Round to the nearest tenth. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

16. SOLUTION: r = 5 yd and h = 18 yd

ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches ANSWER: 3 SOLUTION: 1413.7 yd If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 1534.25 in

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 407.2 cm3

16. SOLUTION: r = 5 yd and h = 18 yd 18.

SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder. ANSWER: 3 1413.7 yd

17. SOLUTION: r = 6 cm and h = 3.6 cm.

Now you can find the volume.

ANSWER:

407.2 cm3 ANSWER: 3 823.0 in

18. SOLUTION: 19. r = 5.5 in. SOLUTION: Use the Pythagorean Theorem to find the height of If two solids have the same height h and the same the cylinder. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm.

Now you can find the volume. ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil

ANSWER: in the planter if the planter is filled to inches 3 823.0 in below the top? SOLUTION:

The planter is to be filled inches below the top, so

r = 7.5 mm and h = 15.2 mm. ANSWER: 3 2740.5 in

20. PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches SOLUTION: The volume of the empty space is the difference of

below the top? volumes of the rectangular prism and the cylinders. SOLUTION:

The planter is to be filled inches below the top, so

ANSWER: 521.5 cm3

22. SANDCASTLES In a sandcastle competition, ANSWER: contestants are allowed to use only water, shovels, 3 2740.5 in and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that 21. SHIPPING A box 18 centimeters by 9 centimeters are 2 feet tall to hold enough sand for one contestant. by 15 centimeters is being used to ship two What should the diameter of the cylinders be? cylindrical candles. Each candle has a diameter of 9 SOLUTION: centimeters and a height of 15 centimeters, as shown

at the right. What is the volume of the empty space 3 in the box? V = 10 ft and h = 2 ft Use the formula to find r.

SOLUTION:

The volume of the empty space is the difference of Therefore, the diameter of the cylinders should be volumes of the rectangular prism and the cylinders. about 2.52 ft.

ANSWER: 2.52 ft Find the volume of the solid formed by each net.

ANSWER: 521.5 cm3

22. SANDCASTLES In a sandcastle competition, 23. contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct SOLUTION: amount of sand, they want to create cylinders that The middle piece of the net is the front of the solid. are 2 feet tall to hold enough sand for one contestant. The top and bottom pieces are the bases and the What should the diameter of the cylinders be? pieces on the ends are the side faces. This is a SOLUTION: triangular prism.

3 One leg of the base 14 cm and the hypotenuse 31.4 V = 10 ft and h = 2 ft Use the formula to find r. cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm.

The volume V of a prism is V = Bh, where B is the

area of the base, h is the height of the prism.

Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER: 2.52 ft Find the volume of the solid formed by each ANSWER: net. 3934.9 cm3

23. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the 24. pieces on the ends are the side faces. This is a triangular prism. SOLUTION: The circular bases at the top and bottom of the net One leg of the base 14 cm and the hypotenuse 31.4 indicate that this is a cylinder. If the middle piece cm. Use the Pythagorean Theorem to find the height were a rectangle, then the prism would be right. of the base. However, since the middle piece is a parallelogram, it is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant The height of the prism is 20 cm. height is 6 m.

The volume V of a prism is V = Bh, where B is the If two solids have the same height h and the same area of the base, h is the height of the prism. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3934.9 cm3

ANSWER: 3 48.9 m

25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the

24. same radius, what is the height of the larger can? SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique. SOLUTION: The radius is 1.8 m, the height is 4.8 m, and the slant The volume of the smaller can is height is 6 m. The volume of the new can is 130% of the smaller If two solids have the same height h and the same can, with the same radius. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The height of the new can will be 35.1 cm.

ANSWER: ANSWER: 35.1 cm 3 48.9 m 26. CHANGING DIMENSIONS A cylinder has a 25. FOOD A cylindrical can of baked potato chips has a radius of 5 centimeters and a height of 8 centimeters. height of 27 centimeters and a radius of 4 Describe how each change affects the volume of the centimeters. A new can is advertised as being 30% cylinder. larger than the regular can. If both cans have the a. The height is tripled. same radius, what is the height of the larger can? b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller a. can, with the same radius. When the height is tripled, h = 3h.

When the height is tripled, the volume is multiplied by 3. The height of the new can will be 35.1 cm. b. When the radius is tripled, r = 3r.

ANSWER: 35.1 cm

d. The dimensions are exchanged. c. When the height and the radius are tripled, r = 3r SOLUTION: and h = 3h.

a. When the height is tripled, h = 3h.

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

Compare to the original volume.

The volume is multiplied by .

So, when the radius is tripled, the volume is multiplied ANSWER: by 9. a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. When the height and the radius are tripled, r = 3r c. The volume is multiplied by 3 or 27. and h = 3h. d. The volume is multiplied by

When the height and the radius are tripled, the a. If the weight of the container with the soil is 20 volume is multiplied by 27. pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming d. When the dimensions are exchanged, r = 8 and h that all other factors are favorable, how well should a = 5 cm. plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION:

Compare to the original volume.

The volume is multiplied by . a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by The weight of the soil is the weight of the pot with 27. SOIL A soil scientist wants to determine the bulk soil minus the weight of the pot. density of a potting soil to assess how well a specific W = 20 – 5 = 15 lbs. plant will grow in it. The density of the soil sample is the ratio of its weight to its The soil density is thus: volume.

a. If the weight of the container with the soil is 20

pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming 3 3 b. 0.0018 lb/in is close to 0.0019 lb/in so the plant that all other factors are favorable, how well should a should grow fairly well. plant grow in this soil if a bulk density of 0.018 pound

per square inch is desirable for root growth? Explain. c. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds?

SOLUTION:

ANSWER: 3 a. 0.0019 lb / in a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round The weight of the soil is the weight of the pot with to the nearest tenth if necessary. soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus:

28. SOLUTION:

3 3 The solid is a combination of two rectangular prisms. b. 0.0018 lb/in is close to 0.0019 lb/in so the plant The base of one rectangular prism is 5 cm by 3 cm should grow fairly well. and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm. c.

ANSWER: ANSWER: 225 cm3 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round 29. to the nearest tenth if necessary. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 120 m

ANSWER: 225 cm3

30. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid

is the sum of the volumes of the two rectangular prisms.

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the

thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the ANSWER: holder? 3 120 m

30. SOLUTION: SOLUTION: The volume of the rubberized material is the The solid is a combination of a rectangular prism and difference between the volumes of the container and two half cylinders. the space used for the can. The container has a

radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

ANSWER:

3 260.5 in Therefore, the volume of the rubberized material is 3 31. MANUFACTURING A can 12 centimeters tall fits about 304.1 cm . into a rubberized cylindrical holder that is 11.5 ANSWER: centimeters tall, including 1 centimeter for the 3 thickness of the base of the holder. The thickness of 304.1 cm the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the Find each measure to the nearest tenth. holder? 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a

radius of and a height of 11.5 cm.

The empty space used to keep the can has a radius ANSWER: of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 5.7 cm 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder. 33. A cylinder has a surface area of 144π square inches

and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

The radius is 6. Find the volume.

ANSWER: 5.7 cm ANSWER: 33. A cylinder has a surface area of 144π square inches 3 and a height of 6 inches. What is the volume? 678.6 in SOLUTION: 34. A rectangular prism has a surface area of 432 Use the surface area formula to solve for r. square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

Find the volume.

The radius is 6. Find the volume.

ANSWER: 3 576 in

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new ANSWER: building. According to the specifications, the diameter 678.6 in3 of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. 34. A rectangular prism has a surface area of 432 What is the difference in volume between the largest square inches, a height of 6 inches, and a width of 12 and smallest possible column? Round to the nearest inches. What is the volume? tenth cubic centimeter. SOLUTION: SOLUTION: Use the surface area formula to find the length of the The volume will be the highest when the diameter is base of the prism. 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

Find the volume.

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular ANSWER: swimming pool is sloped so one end of the pool is 6 3 576 in feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume 35. ARCHITECTURE A cylindrical stainless steel of water it takes to fill the pool. column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest SOLUTION: tenth cubic centimeter. The swimming pool is a combination of a rectangular SOLUTION: prism and a trapezoidal prism. The base of the The volume will be the highest when the diameter is rectangular prism is 6 ft by 10 ft and the height is 15 95 cm and will be the lowest when it is 30 cm.That is ft. The bases of the trapezoidal prism are 6 ft and 3 when the radii are 47.5 cm and 15 cm respectively. ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of Find the difference between the volumes. the solid is the sum of the volumes of the two prisms.

ANSWER: ANSWER: 12-4 Volumes of Prisms and Cylinders 3 3,190,680.0 cm3 1575 ft

36. CCSS MODELING The base of a rectangular 37. CHANGING DIMENSIONS A soy milk company swimming pool is sloped so one end of the pool is 6 is planning a promotion in which the volume of soy feet deep and the other end is 3 feet deep, as shown milk in each container will be increased by 25%. The in the figure. If the width is 15 feet, find the volume company wants the base of the container to stay the of water it takes to fill the pool. same. What will be the height of the new containers?

SOLUTION:

The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the SOLUTION: rectangular prism is 6 ft by 10 ft and the height is 15 Find the volume of the original container. ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms. The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

ANSWER: 3 1575 ft ANSWER: 37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the 38. DESIGN Sketch and label (in inches) three different same. What will be the height of the new containers? designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume: SOLUTION: Find the volume of the original container.

eSolutionsThe Manualvolume- Powered of the newby Cognero container is 125% of the Page 13 original container, with the same base dimensions. The last equation gives us a relation between the Use 1.25V and B to find h. radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height.

ANSWER:

If we choose a height of say 4 in., then we can solve for the radius. 38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and The last equation gives us a relation between the we can solve for the third. Let l = 2.25 in. and w = radius and height of the cylinder that must be fulfilled 2.5 in. to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height.

If we choose a height of say 4 in., then we can solve for the radius.

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in. ANSWER: Sample answers:

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

ANSWER: Sample answers:

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio 39. Find the volume of the regular pentagonal prism by and $700 for removal of the old patio. Which is the dividing it into five equal triangular prisms. Describe less expensive option? Explain. the base area and height of each triangular prism. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd SOLUTION: Find the volume. The base of the prism can be divided into 5

congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 The total cost for the second contractor is about 1100 cm ; Each triangular prism has a base area of . 2 or 22 cm and a height of 10 cm. Therefore, the second contractor is a less expensive option. 40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 ANSWER: feet long, 12 feet wide, and 4 inches thick. One Because 2.96 yd3 of concrete are needed, the contractor bid $2225 for the project. A second second contractor is less expensive at $2181.50. contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the 41. MULTIPLE REPRESENTATIONS In this less expensive option? Explain. problem, you will investigate right and oblique SOLUTION: cylinders . Convert all of the dimensions to yards. a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a

20 feet = yd diameter of 6 meters. 12 feet = 4 yd 4 in. = yd b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume Find the volume. greater than, less than, or equal to the volume of the cylinder? Explain.

The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option.

ANSWER: b. Find the volume of each. Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume The volume of the square prism is greater. greater than, less than, or equal to the volume of c. Do each scenario. the cylinder? Explain.

c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain.

SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

2 For example, if x = 0.5, then x = 0.25, which is less b. Find the volume of each. than x.

ANSWER: a.

b. Greater than; a square with a side length of 6 m 2 The volume of the square prism is greater. has an area of 36 m . A circle with a diameter of 6 c. Do each scenario. 2 m has an area of 9π or 28.3 m . Since the heights are the same, the volume of the square prism is greater.

42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 Assuming x > 1, multiplying the radius by x makes units. Is either of them correct? Explain your 2 the volume x times greater. reasoning.

For example, if x = 0.5, then x2 = 0.25, which is less than x.

ANSWER: a. SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base. b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 2 m has an area of 9π or 28.3 m . Since the heights are the same, the volume of the square prism is

greater. ANSWER:

c. Multiplying the radius by x ; since the volume is Francisco; Valerie incorrectly used as the represented by π r 2 h, multiplying the height by x length of one side of the triangular base. Francisco

makes the volume x times greater. Multiplying the used a different approach, but his solution is correct. radius by x makes the volume x 2 times greater, 43. CHALLENGE A cylindrical can is used to fill a assuming x > 1. container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the 42. CCSS CRITIQUE Franciso and Valerie each container if it is each of the following shapes. calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 a. rectangular prism units. Is either of them correct? Explain your

reasoning. b. square prism

c. triangular prism with a right triangle as the base

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct. SOLUTION:

Francisco used the standard formula for the volume 3 of a solid, V = Bh. The area of the base, B, is one- The volume of the can is 20π in . It takes three full half the apothem multiplied by the perimeter of the cans to fill the container, so the volume of the base. 3 container is 60π in .

a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5.

ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

43. CHALLENGE A cylindrical can is used to fill a 3 by 5 by 4π container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes. b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. a. rectangular prism

b. square prism

c. triangular prism with a right triangle as the base

5 by 5 by

SOLUTION: c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. 3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the container is 60π in 3.

a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5.

3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π 3 by 5 by 4π

b. Choose some basic values for 2 of the sides, and b. 5 by 5 by then determine the third side. Base: 5 by 5. c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of 5 by 5 by soil? How do I know what to order?

SOLUTION: c. Choose some basic values for 2 of the sides, and Sample answer: The nursery means a cubic yard, then determine the third side. Base: Legs: 3 by 4. 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine 3 by 4 by 10π the number of cubic yards of soil needed.

ANSWER: 45. OPEN ENDED Draw and label a prism that has a Sample answers: volume of 50 cubic centimeters. SOLUTION: a. π 3 by 5 by 4 Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest

b. 5 by 5 by values to choose.

Sample answer: c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum.

I am new to gardening. The nursery will deliver a ANSWER: truckload of soil, which they say is 4 yards. I Sample answer: know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine 46. REASONING Determine whether the following the number of cubic yards of soil needed. statement is true or false. Explain. Two cylinders with the same height and the same ANSWER: lateral area must have the same volume. Sample answer: The nursery means a cubic yard, 3 SOLUTION: which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine True; if two cylinders have the same height (h1 = h2) the number of cubic yards of soil needed. and the same lateral area (L1 = L2), the circular 45. OPEN ENDED Draw and label a prism that has a bases must have the same area. volume of 50 cubic centimeters. SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest values to choose.

Sample answer:

The radii must also be equal.

ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 ANSWER: same area. Therefore, πr h is the same for each Sample answer: cylinder. 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION:

Both formulas involve multiplying the area of the 46. REASONING Determine whether the following base by the height. The base of a prism is a polygon, statement is true or false. Explain. so the expression representing the area varies, Two cylinders with the same height and the same depending on the type of polygon it is. The base of a lateral area must have the same volume. cylinder is a circle, so its area is πr2.

SOLUTION: ANSWER: True; if two cylinders have the same height (h1 = h2) Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is and the same lateral area (L1 = L2), the circular a polygon, so the expression representing the area bases must have the same area. varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr .

48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm The radii must also be equal. C 17 cm D 11.5 cm ANSWER: True; if two cylinders have the same height and the SOLUTION: same lateral area, the circular bases must have the 2 same area. Therefore, πr h is the same for each cylinder.

47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, ANSWER: depending on the type of polygon it is. The base of a B cylinder is a circle, so its area is πr2. 49. A cylindrical tank used for oil storage has a height ANSWER: that is half the length of its radius. If the volume of Sample answer: Both formulas involve multiplying the the tank is 1,122,360 ft3, what is the tank’s radius? area of the base by the height. The base of a prism is F 89.4 ft a polygon, so the expression representing the area G 178.8 ft varies, depending on the type of polygon it is. The H 280.9 ft 2 base of a cylinder is a circle, so its area is πr . J 561.8 ft 48. The volume of a triangular prism is 1380 cubic SOLUTION: centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm C 17 cm D 11.5 cm SOLUTION:

ANSWER: F

SHORT RESPONSE 50. What is the ratio of the area of the circle to the area of the square? ANSWER: B 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of SOLUTION: 3 the tank is 1,122,360 ft , what is the tank’s radius? The radius of the circle is 2x and the length of each F 89.4 ft side of the square is 4x. So, the ratio of the areas can G 178.8 ft be written as shown. H 280.9 ft J 561.8 ft SOLUTION: ANSWER:

SOLUTION: ANSWER: Find the 0.5% of $85,000. F

50. SHORT RESPONSE What is the ratio of the area Therefore, the correct choice is C. of the circle to the area of the square? ANSWER: C

Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if SOLUTION: necessary. The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

52.

ANSWER: SOLUTION: The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of 51. SAT/ACT A county proposes to enact a new 0.5% the base. property tax. What would be the additional tax amount for a landowner whose property has a The slant height is the height of each of the taxable value of $85,000? congruent lateral triangular faces. Use the A $4.25 Pythagorean Theorem to find the slant height. B $170 C $425 D $4250 E $42,500 SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C

Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem SOLUTION: to find the height h of the triangle. The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

The perimeter is P = 3 × 10 or 30 feet.

So, the area of the base B is ft2.

Find the perimeter and area of the equilateral Find the lateral area L and surface area S of the triangle for the base. Use the Pythagorean Theorem regular pyramid. to find the height h of the triangle.

So, the lateral area of the pyramid is about 212.1 ft2.

Therefore, the surface area of the pyramid is about 2 The perimeter is P = 3 × 10 or 30 feet. 255.4 ft . ANSWER: 212.1 ft2; 255.4 ft2

So, the area of the base B is ft2. 53. Find the lateral area L and surface area S of the SOLUTION: regular pyramid. The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the lateral area of the pyramid is about 212.1 ft2.

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular pyramid is , where Therefore, the surface area of the pyramid is about L is the lateral area and B is the area of the base. 2 255.4 ft .

ANSWER: 212.1 ft2; 255.4 ft2

Therefore, the surface area of the pyramid is 175 cm2.

53. ANSWER: 2 2 SOLUTION: 126 cm ; 175 cm

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant 54. height is 9 cm. SOLUTION: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

Use a trigonometric ratio to find the measure of the apothem a.

Therefore, the surface area of the pyramid is 175 cm2. ANSWER: 2 2 126 cm ; 175 cm Find the lateral area and surface area of the pyramid.

54. SOLUTION: The pyramid has a slant height of 15 inches and the So, the lateral area of the pyramid is 472.5 in2. base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

Therefore, the surface area of the pyramid is about 758.9 in2.

Use a trigonometric ratio to find the measure of the ANSWER: apothem a. 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

Find the lateral area and surface area of the pyramid. SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in . So, the lateral area of the pyramid is 472.5 in2. ANSWER:

205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION: SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the The diameter of the circle is about 11.4 m. diameter. SOLUTION: ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. ANSWER: 7.8 ft SOLUTION: 59. Find the radius of a circle with an area of 271 square inches. SOLUTION:

The diameter of the circle is about 11.4 m.

ANSWER:

11.4 cm

58. The area of a circle is 191 square feet. Find the radius. ANSWER: 9.3 in. SOLUTION: Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the ANSWER: lengths of its diagonals, d1 and d2. 7.8 ft d1 = 12 in. and d2 = 7 + 13 = 20 in. 59. Find the radius of a circle with an area of 271 square

inches. SOLUTION:

ANSWER: 2 120 in

ANSWER: 9.3 in. 61. Find the area of each trapezoid, rhombus, or SOLUTION: kite. The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2. ANSWER: d = 12 in. and d = 7 + 13 = 20 in. 2 1 2 378 m

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right Find the volume of each prism.

1. SOLUTION: SOLUTION: If two solids have the same height h and the same The volume V of a prism is V = Bh, where B is the cross-sectional area B at every level, then they have area of a base and h is the height of the prism. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

3 The volume is 108 cm .

ANSWER: ANSWER: 3 108 cm3 26.95 m 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 2. an oblique one of the same height and cross sectional area are same. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 3 396 in 5. 3. the oblique rectangular prism shown at the right SOLUTION:

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 206.4 ft3 area are same.

6. SOLUTION: ANSWER: If two solids have the same height h and the same cross-sectional area B at every level, then they have 3 26.95 m the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross 4. an oblique pentagonal prism with a base area of 42 sectional area are same. square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters

SOLUTION: ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 1025.4 cm3 5. 8. a cylinder with a radius of 4.2 inches and a height of SOLUTION: 7.4 inches SOLUTION:

ANSWER: ANSWER: 3 206.4 ft 3 410.1 in

9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the 6. lap pool? SOLUTION: A 4000 If two solids have the same height h and the same B 6400 cross-sectional area B at every level, then they have C 30,000 the same volume. So, the volume of a right cylinder D 48,000 and an oblique one of the same height and cross sectional area are same. SOLUTION:

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

ANSWER: Therefore, the correct choice is D. 3 1357.2 m ANSWER: 7. a cylinder with a diameter of 16 centimeters and a D height of 5.1 centimeters CCSS SENSE-MAKING Find the volume of SOLUTION: each prism.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. ANSWER: The height of the prism is 5 in. 3 1025.4 cm

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches

SOLUTION:

ANSWER: 30 in3

ANSWER: 3 410.1 in 11. 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to SOLUTION: be filled to four feet deep and each cubic foot holds The base is a triangle with a base length of 11 m and 7.5 gallons, how many gallons will it take to fill the the corresponding height of 7 m. The height of the lap pool? prism is 14 m. A 4000 B 6400 C 30,000 D 48,000

SOLUTION: ANSWER: 3 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. 12.

ANSWER: SOLUTION: D The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. CCSS SENSE-MAKING Find the volume of each prism. Use the Pythagorean Theorem to find the height of the base.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

The height of the prism is 6 cm.

ANSWER: 30 in3 ANSWER: 324 cm3

11. SOLUTION: The base is a triangle with a base length of 11 m and 13. the corresponding height of 7 m. The height of the prism is 14 m. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: The volume V of a prism is V = Bh, where B is the 3 area of a base and h is the height of the prism. 539 m 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

ANSWER: 12. 58.14 ft3

SOLUTION: 14. an oblique hexagonal prism with a height of 15 The base is a right triangle with a leg length of 9 cm centimeters and with a base area of 136 square and the hypotenuse of length 15 cm. centimeters

The height of the prism is 6 cm. ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches ANSWER: SOLUTION: 3 If two solids have the same height h and the same 324 cm cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

13. SOLUTION:

If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and 3 1534.25 in an oblique one of the same height and cross sectional area are same. CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is 16.

SOLUTION: ANSWER: r = 5 yd and h = 18 yd 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: ANSWER: If two solids have the same height h and the same 3 cross-sectional area B at every level, then they have 1413.7 yd the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches

SOLUTION: ANSWER: If two solids have the same height h and the same 407.2 cm3 cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

18. SOLUTION: r = 5.5 in.

ANSWER: Use the Pythagorean Theorem to find the height of the cylinder. 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. SOLUTION: r = 5 yd and h = 18 yd

Now you can find the volume.

ANSWER: 3 1413.7 yd

ANSWER: 3 823.0 in 17. SOLUTION: r = 6 cm and h = 3.6 cm. 19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have

the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same.

3 407.2 cm r = 7.5 mm and h = 15.2 mm.

18.

SOLUTION: r = 5.5 in. ANSWER: 2686.1 mm3 Use the Pythagorean Theorem to find the height of the cylinder. 20. PLANTER A planter is in the shape of a

rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER: ANSWER: 3 3 2740.5 in 823.0 in 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown 19. at the right. What is the volume of the empty space in the box? SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional

area are same.

r = 7.5 mm and h = 15.2 mm. SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

ANSWER: 3 2686.1 mm

20. PLANTER A planter is in the shape of a ANSWER: 3 rectangular prism 18 inches long, inches deep, 521.5 cm

and 12 inches high. What is the volume of potting soil 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, in the planter if the planter is filled to inches and 10 cubic feet of sand. To transport the correct below the top? amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. SOLUTION: What should the diameter of the cylinders be? The planter is to be filled inches below the top, so SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3 2740.5 in Therefore, the diameter of the cylinders should be about 2.52 ft. 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two ANSWER: cylindrical candles. Each candle has a diameter of 9 2.52 ft centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space Find the volume of the solid formed by each in the box? net.

23. SOLUTION: The volume of the empty space is the difference of SOLUTION: volumes of the rectangular prism and the cylinders. The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

ANSWER: 3 521.5 cm The height of the prism is 20 cm.

22. SANDCASTLES In a sandcastle competition, The volume V of a prism is V = Bh, where B is the contestants are allowed to use only water, shovels, area of the base, h is the height of the prism. and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION: ANSWER: 3 V = 10 ft and h = 2 ft Use the formula to find r. 3934.9 cm3

Therefore, the diameter of the cylinders should be

about 2.52 ft. 24. ANSWER: SOLUTION: 2.52 ft The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece Find the volume of the solid formed by each were a rectangle, then the prism would be right. net. However, since the middle piece is a parallelogram, it is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

If two solids have the same height h and the same 23. cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional The middle piece of the net is the front of the solid. area are same. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

ANSWER: The height of the prism is 20 cm. 3 48.9 m

The volume V of a prism is V = Bh, where B is the 25. FOOD A cylindrical can of baked potato chips has a area of the base, h is the height of the prism. height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

ANSWER: 3934.9 cm3 SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

ANSWER: The radius is 1.8 m, the height is 4.8 m, and the slant 35.1 cm height is 6 m. 26. CHANGING DIMENSIONS A cylinder has a If two solids have the same height h and the same radius of 5 centimeters and a height of 8 centimeters. cross-sectional area B at every level, then they have Describe how each change affects the volume of the the same volume. So, the volume of a right prism and cylinder. an oblique one of the same height and cross sectional a. The height is tripled. area are same. b. The radius is tripled.

c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

ANSWER: 3 48.9 m a. When the height is tripled, h = 3h. 25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

When the height is tripled, the volume is multiplied by 3.

b. When the radius is tripled, r = 3r. SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the When the height and the radius are tripled, the cylinder. volume is multiplied by 27. a. The height is tripled. b. The radius is tripled. d. When the dimensions are exchanged, r = 8 and h c. Both the radius and the height are tripled. = 5 cm. d. The dimensions are exchanged. SOLUTION:

Compare to the original volume.

When the height is tripled, h = 3h. The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. c. 3 The volume is multiplied by 3 or 27.

When the height is tripled, the volume is multiplied by d. The volume is multiplied by 3. b. When the radius is tripled, r = 3r. 27. SOIL A soil scientist wants to determine the bulk

density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming So, when the radius is tripled, the volume is multiplied that all other factors are favorable, how well should a by 9. plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. When the height and the radius are tripled, r = 3r c. If a bag of this soil holds 2.5 cubic feet, what is its and h = 3h. weight in pounds? SOLUTION:

When the height and the radius are tripled, the a. First calculate the volume of soil in the pot. Then volume is multiplied by 27. divide the weight of the soil by the volume.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs. Compare to the original volume. The soil density is thus:

The volume is multiplied by .

ANSWER: b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant a. The volume is multiplied by 3. should grow fairly well.

b. 2 The volume is multiplied by 3 or 9. c 3 . c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its ANSWER: volume. 3 a. 0.0019 lb / in a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 b. The plant should grow well in this soil since the pounds, what is the soil’s bulk density? b. Assuming bulk density of 0.0019 lb / in3 is close to the desired that all other factors are favorable, how well should a 3 bulk density of 0.0018 lb / in . plant grow in this soil if a bulk density of 0.018 pound

per square inch is desirable for root growth? Explain. c. 8.3 lb c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? Find the volume of each composite solid. Round SOLUTION: to the nearest tenth if necessary.

28.

a. First calculate the volume of soil in the pot. Then SOLUTION: divide the weight of the soil by the volume. The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

ANSWER: The soil density is thus: 225 cm3

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

29.

SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms. ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round ANSWER: to the nearest tenth if necessary. 3 120 m

28. 30. SOLUTION: The solid is a combination of two rectangular prisms. SOLUTION: The base of one rectangular prism is 5 cm by 3 cm The solid is a combination of a rectangular prism and and the height is 11 cm. The base of the other prism two half cylinders. is 4 cm by 3 cm and the height is 5 cm.

ANSWER: 225 cm3 ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the 29. holder? SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a

radius of and a height of 11.5 cm.

The empty space used to keep the can has a radius ANSWER: of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 3 2 120 m volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

30. Therefore, the volume of the rubberized material is 3 SOLUTION: about 304.1 cm . The solid is a combination of a rectangular prism and ANSWER: two half cylinders. 3 304.1 cm Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

ANSWER: 260.5 in3

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume. Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth.

32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? ANSWER: SOLUTION: 678.6 in3

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? Find the volume. SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

Find the difference between the volumes.

ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432

square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: ANSWER: 3 Use the surface area formula to find the length of the 3,190,680.0 cm base of the prism. 36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

Find the volume.

and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: ANSWER: 3 The volume will be the highest when the diameter is 1575 ft 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively. 37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy Find the difference between the volumes. milk in each container will be increased by 25%. The company wants the base of the container to stay the same. What will be the height of the new containers?

ANSWER: SOLUTION: 3 Find the volume of the original container. 3,190,680.0 cm 36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool. The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ANSWER: ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms. 38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume: ANSWER: 3 1575 ft

37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the

same. What will be the height of the new containers? The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height.

SOLUTION: Find the volume of the original container.

If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

ANSWER: For any rectangular container, the volume equation is:

38. DESIGN Sketch and label (in inches) three different Choose numbers for any two of the dimensions and designs for a dry ingredient measuring cup that holds we can solve for the third. Let l = 2.25 in. and w = 1 cup. Be sure to include the dimensions in each 2.5 in. drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers: 12-4 Volumes of Prisms and Cylinders For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by eSolutions Manual - Powered by Cognero dividing it into five equal triangular prisms. DescribePage 14 the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: ANSWER: Sample answers: 3 1100 cm ; Each triangular prism has a base area of

2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe

the base area and height of each triangular prism.

The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive SOLUTION: option. The base of the prism can be divided into 5 ANSWER: congruent triangles of a base 8 cm and the 3 corresponding height 5.5 cm. So, the pentagonal Because 2.96 yd of concrete are needed, the prism is a combination of 5 triangular prisms of height second contractor is less expensive at $2181.50. 10 cm. Find the base area of each triangular prism. 41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders Therefore, the volume of the pentagonal prism is .

a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a ANSWER: diameter of 6 meters. 3 1100 cm ; Each triangular prism has a base area of 2 b. VERBAL A square prism has a height of 10 or 22 cm and a height of 10 cm. meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of 40. PATIOS Mr. Thomas is planning to remove an old the cylinder? Explain. patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One c. ANALYTICAL Describe which change affects contractor bid $2225 for the project. A second the volume of the cylinder more: multiplying the contractor bid $500 per cubic yard for the new patio height by x or multiplying the radius by x. Explain. and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: a. The oblique cylinder should look like the right SOLUTION: cylinder (same height and size), except that it is Convert all of the dimensions to yards. pushed a little to the side, like a slinky.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume. b. Find the volume of each.

The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive The volume of the square prism is greater. option. c. Do each scenario. ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

Assuming x > 1, multiplying the radius by x makes b. VERBAL A square prism has a height of 10 2 meters and a base edge of 6 meters. Is its volume the volume x times greater. greater than, less than, or equal to the volume of 2 the cylinder? Explain. For example, if x = 0.5, then x = 0.25, which is less than x. c. ANALYTICAL Describe which change affects ANSWER: the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. a. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 2 m has an area of 9π or 28.3 m . Since the heights b. Find the volume of each. are the same, the volume of the square prism is

greater.

42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning. The volume of the square prism is greater. c. Do each scenario.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base. Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less than x. ANSWER: ANSWER: Francisco; Valerie incorrectly used as the a. length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

a. rectangular prism b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 b. square prism m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is c. triangular prism with a right triangle as the base greater.

3 42. CCSS CRITIQUE Franciso and Valerie each The volume of the can is 20π in . It takes three full calculated the volume of an equilateral triangular cans to fill the container, so the volume of the prism with an apothem of 4 units and height of 5 container is 60π in 3. units. Is either of them correct? Explain your reasoning. a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct. 3 by 5 by 4π

Francisco used the standard formula for the volume b. Choose some basic values for 2 of the sides, and of a solid, V = Bh. The area of the base, B, is one- then determine the third side. Base: 5 by 5. half the apothem multiplied by the perimeter of the base.

ANSWER: Francisco; Valerie incorrectly used as the

length of one side of the triangular base. Francisco used a different approach, but his solution is correct. 5 by 5 by

43. CHALLENGE A cylindrical can is used to fill a c. container with liquid. It takes three full cans to fill the Choose some basic values for 2 of the sides, and

container. Describe possible dimensions of the then determine the third side. Base: Legs: 3 by 4. container if it is each of the following shapes.

a. rectangular prism

b. square prism

c. triangular prism with a right triangle as the base 3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π SOLUTION:

3 b. 5 by 5 by The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the c. base with legs measuring 3 in. and 4 in., height 3 container is 60π in . 10π in.

a. Choose some basic values for 2 of the sides, and 44. WRITING IN MATH Write a helpful response to then determine the third side. Base: 3 by 5. the following question posted on an Internet gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order?

SOLUTION: 3 by 5 by 4π Sample answer: The nursery means a cubic yard, 3 b. Choose some basic values for 2 of the sides, and which is 3 or 27 cubic feet. Find the volume of your then determine the third side. Base: 5 by 5. garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. 5 by 5 by SOLUTION: Choose 3 values that multiply to make 50. The c. Choose some basic values for 2 of the sides, and factors of 50 are 2, 5, 5, so these are the simplest then determine the third side. Base: Legs: 3 by 4. values to choose.

Sample answer:

3 by 4 by 10π ANSWER: ANSWER: Sample answer: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by 46. REASONING Determine whether the following c. base with legs measuring 3 in. and 4 in., height statement is true or false. Explain. 10π in. Two cylinders with the same height and the same lateral area must have the same volume. 44. WRITING IN MATH Write a helpful response to SOLUTION: the following question posted on an Internet True; if two cylinders have the same height (h = h ) gardening forum. 1 2 and the same lateral area (L1 = L2), the circular I am new to gardening. The nursery will deliver a bases must have the same area. truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. The radii must also be equal. ANSWER: Sample answer: The nursery means a cubic yard, ANSWER: which is 33 or 27 cubic feet. Find the volume of your True; if two cylinders have the same height and the garden in cubic feet and divide by 27 to determine same lateral area, the circular bases must have the 2 the number of cubic yards of soil needed. same area. Therefore, πr h is the same for each cylinder. 45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder SOLUTION: similar? How are they different? Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest SOLUTION: values to choose. Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, Sample answer: so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is ANSWER: a polygon, so the expression representing the area Sample answer: varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr .

48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? 46. REASONING Determine whether the following A 34.5 cm statement is true or false. Explain. B 23 cm Two cylinders with the same height and the same C 17 cm lateral area must have the same volume. D 11.5 cm SOLUTION: SOLUTION:

True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: B The radii must also be equal.

ANSWER: 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of True; if two cylinders have the same height and the 3 same lateral area, the circular bases must have the the tank is 1,122,360 ft , what is the tank’s radius? 2 F 89.4 ft same area. Therefore, πr h is the same for each G cylinder. 178.8 ft H 280.9 ft 47. WRITING IN MATH How are the formulas for J 561.8 ft the volume of a prism and the volume of a cylinder SOLUTION: similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr . ANSWER: 48. The volume of a triangular prism is 1380 cubic F centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is 50. SHORT RESPONSE What is the ratio of the area the height of the prism? of the circle to the area of the square? A 34.5 cm B 23 cm C 17 cm D 11.5 cm SOLUTION: SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% ANSWER: property tax. What would be the additional tax B amount for a landowner whose property has a taxable value of $85,000? 49. A cylindrical tank used for oil storage has a height A $4.25 that is half the length of its radius. If the volume of B $170 3 the tank is 1,122,360 ft , what is the tank’s radius? C $425 F 89.4 ft D $4250 G 178.8 ft E $42,500 H 280.9 ft SOLUTION: J 561.8 ft Find the 0.5% of $85,000. SOLUTION:

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52.

ANSWER: SOLUTION: F The lateral area L of a regular pyramid is , 50. SHORT RESPONSE What is the ratio of the area where is the slant height and P is the perimeter of of the circle to the area of the square? the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

SAT/ACT 51. A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 D $4250 E $42,500 Find the perimeter and area of the equilateral SOLUTION: triangle for the base. Use the Pythagorean Theorem Find the 0.5% of $85,000. to find the height h of the triangle.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of The perimeter is P = 3 × 10 or 30 feet. the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

ANSWER: 212.1 ft2; 255.4 ft2

53. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of The perimeter is P = 3 × 10 or 30 feet. the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the area of the base B is ft2. So, the lateral area of the pyramid is 126 cm2.

Find the lateral area L and surface area S of the regular pyramid. The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is about 212.1 ft2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 2 255.4 ft .

53. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm. Use a trigonometric ratio to find the measure of the apothem a.

2 So, the lateral area of the pyramid is 126 cm .

Find the lateral area and surface area of the The surface area S of a regular pyramid. pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 758.9 in2.

54. ANSWER: SOLUTION: 472.5 in2; 758.9 in2 The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. 55. BAKING Many baking pans are given a special A central angle of the hexagon is or 60°, so the nonstick coating. A rectangular cake pan is 9 inches angle formed in the triangle below is 30°. by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 Use a trigonometric ratio to find the measure of the 9)(2) + 13(9) = 205 in . apothem a. ANSWER:

205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. Find the lateral area and surface area of the SOLUTION: pyramid.

So, the lateral area of the pyramid is 472.5 in2.

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. Therefore, the surface area of the pyramid is about SOLUTION: 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

ANSWER: 11.4 cm SOLUTION: 58. The area of a circle is 191 square feet. Find the The area that needs to be coated is the sum of the radius. lateral area and one base area. SOLUTION: Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION: ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. The diameter of the circle is about 11.4 m. SOLUTION: The area A of a kite is one half the product of the ANSWER: lengths of its diagonals, d and d . 11.4 cm 1 2

58. The area of a circle is 191 square feet. Find the d1 = 12 in. and d2 = 7 + 13 = 20 in.

radius. SOLUTION:

ANSWER: 2 120 in

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square 61. inches. SOLUTION: SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 ANSWER: 378 m 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

Find the volume of each prism.

2. SOLUTION: 1. The volume V of a prism is V = Bh, where B is the

SOLUTION: area of a base and h is the height of the prism.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm . ANSWER: 3 ANSWER: 396 in 3 108 cm 3. the oblique rectangular prism shown at the right

ANSWER:

3 26.95 m

ANSWER: 4. an oblique pentagonal prism with a base area of 42 3 square centimeters and a height of 5.2 centimeters 396 in SOLUTION: 3. the oblique rectangular prism shown at the right If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 26.95 m3 5. SOLUTION: 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER:

206.4 ft3

5. SOLUTION:

ANSWER: 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a ANSWER: height of 5.1 centimeters 3 206.4 ft SOLUTION:

ANSWER: ANSWER: 3 3 1357.2 m 410.1 in

7. a cylinder with a diameter of 16 centimeters and a 9. MULTIPLE CHOICE A rectangular lap pool height of 5.1 centimeters measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds SOLUTION: 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000 D 48,000

ANSWER: SOLUTION: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches Each cubic foot holds 7.5 gallons of water. So, the SOLUTION: amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: D ANSWER: CCSS SENSE-MAKING Find the volume of 3 410.1 in each prism.

9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 10. 7.5 gallons, how many gallons will it take to fill the lap pool? SOLUTION: A 4000 The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. B 6400

C 30,000 D 48,000

SOLUTION:

ANSWER: 3 30 in Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: 11. D SOLUTION: CCSS SENSE-MAKING Find the volume of The base is a triangle with a base length of 11 m and each prism. the corresponding height of 7 m. The height of the prism is 14 m.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. ANSWER:

3 539 m

ANSWER: 30 in3 12. SOLUTION: The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm.

Use the Pythagorean Theorem to find the height of 11. the base.

SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the prism is 14 m.

The height of the prism is 6 cm.

ANSWER: 3 539 m

ANSWER: 324 cm3 12. SOLUTION: The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm.

13. Use the Pythagorean Theorem to find the height of the base. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 The height of the prism is 6 cm. B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

ANSWER: 3 58.14 ft ANSWER: 14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square 324 cm3 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 13. area are same. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: 3 The volume V of a prism is V = Bh, where B is the 2040 cm area of a base and h is the height of the prism. 15. a square prism with a base edge of 9.5 inches and a 2 height of 17 inches B = 11.4 ft and h = 5.1 ft. Therefore, the volume is SOLUTION: If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 58.14 ft3 the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 14. an oblique hexagonal prism with a height of 15 area are same. centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 3 area are same. 1534.25 in

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 16. 3 2040 cm SOLUTION: 15. a square prism with a base edge of 9.5 inches and a r = 5 yd and h = 18 yd height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 3 1413.7 yd

17. ANSWER: SOLUTION: 3 1534.25 in r = 6 cm and h = 3.6 cm.

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. ANSWER: SOLUTION: 407.2 cm3 r = 5 yd and h = 18 yd

18. ANSWER: SOLUTION: 3 1413.7 yd r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: Now you can find the volume.

407.2 cm3

18. ANSWER: 3 SOLUTION: 823.0 in r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

r = 7.5 mm and h = 15.2 mm.

Now you can find the volume.

ANSWER: 2686.1 mm3

cross-sectional area B at every level, then they have The planter is to be filled inches below the top, so the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm.

ANSWER: 3 2740.5 in ANSWER: 21. SHIPPING A box 18 centimeters by 9 centimeters 3 2686.1 mm by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 20. PLANTER A planter is in the shape of a centimeters and a height of 15 centimeters, as shown rectangular prism 18 inches long, inches deep, at the right. What is the volume of the empty space in the box? and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

ANSWER: 3 2740.5 in ANSWER: 521.5 cm3 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two 22. SANDCASTLES In a sandcastle competition, cylindrical candles. Each candle has a diameter of 9 contestants are allowed to use only water, shovels, centimeters and a height of 15 centimeters, as shown and 10 cubic feet of sand. To transport the correct at the right. What is the volume of the empty space amount of sand, they want to create cylinders that in the box? are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER: 2.52 ft ANSWER: Find the volume of the solid formed by each 521.5 cm3 net. 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? 23. SOLUTION: SOLUTION: 3 V = 10 ft and h = 2 ft Use the formula to find r. The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

Therefore, the diameter of the cylinders should be about 2.52 ft. The height of the prism is 20 cm.

ANSWER: The volume V of a prism is V = Bh, where B is the 2.52 ft area of the base, h is the height of the prism.

Find the volume of the solid formed by each net.

ANSWER: 3 24. 48.9 m

SOLUTION: 25. FOOD A cylindrical can of baked potato chips has a The circular bases at the top and bottom of the net height of 27 centimeters and a radius of 4 indicate that this is a cylinder. If the middle piece centimeters. A new can is advertised as being 30% were a rectangle, then the prism would be right. larger than the regular can. If both cans have the However, since the middle piece is a parallelogram, it same radius, what is the height of the larger can? is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

area are same. The volume of the new can is 130% of the smaller can, with the same radius.

ANSWER: 3 48.9 m The height of the new can will be 35.1 cm. 25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 ANSWER: centimeters. A new can is advertised as being 30% 35.1 cm larger than the regular can. If both cans have the same radius, what is the height of the larger can? 26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. SOLUTION: c. Both the radius and the height are tripled. The volume of the smaller can is d. The dimensions are exchanged. SOLUTION: The volume of the new can is 130% of the smaller can, with the same radius.

a. When the height is tripled, h = 3h.

The height of the new can will be 35.1 cm.

ANSWER:

35.1 cm When the height is tripled, the volume is multiplied by 26. CHANGING DIMENSIONS A cylinder has a 3. radius of 5 centimeters and a height of 8 centimeters. b. When the radius is tripled, r = 3r. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged.

SOLUTION: So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

a. When the height is tripled, h = 3h.

When the height and the radius are tripled, the volume is multiplied by 27. When the height is tripled, the volume is multiplied by

3. d. When the dimensions are exchanged, r = 8 and h b. When the radius is tripled, r = 3r. = 5 cm.

Compare to the original volume.

So, when the radius is tripled, the volume is multiplied by 9.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk When the height and the radius are tripled, the density of a potting soil to assess how well a specific

volume is multiplied by 27. plant will grow in it. The density of the soil sample is the ratio of its weight to its d. When the dimensions are exchanged, r = 8 and h volume. = 5 cm. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound

per square inch is desirable for root growth? Explain. c. Compare to the original volume. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION:

The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 a. First calculate the volume of soil in the pot. Then c. The volume is multiplied by 3 or 27. divide the weight of the soil by the volume.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume. The weight of the soil is the weight of the pot with soil minus the weight of the pot. a. If the weight of the container with the soil is 20 W = 20 – 5 = 15 lbs. pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming The soil density is thus: that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant SOLUTION: should grow fairly well.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the

bulk density of 0.0019 lb / in3 is close to the desired 3 The weight of the soil is the weight of the pot with bulk density of 0.0018 lb / in . soil minus the weight of the pot. W = 20 – 5 = 15 lbs. c. 8.3 lb

The soil density is thus: Find the volume of each composite solid. Round to the nearest tenth if necessary.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the 3 ANSWER: bulk density of 0.0019 lb / in is close to the desired 3 3 225 cm bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

29.

28. SOLUTION: The solid is a combination of a rectangular prism and SOLUTION: a right triangular prism. The total volume of the solid The solid is a combination of two rectangular prisms. is the sum of the volumes of the two rectangular The base of one rectangular prism is 5 cm by 3 cm prisms. and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: 225 cm3 ANSWER: 3 120 m

30. SOLUTION: 29. The solid is a combination of a rectangular prism and SOLUTION: two half cylinders. The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits ANSWER: into a rubberized cylindrical holder that is 11.5 3 120 m centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

30. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm.

The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 ANSWER: volume V of a cylinder is V = πr h, where r is the 260.5 in3 radius of the base and h is the height of the cylinder.

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the Therefore, the volume of the rubberized material is volume of the rubberized material that makes up the about 304.1 cm3. holder? ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height?

ANSWER: 5.7 cm

Therefore, the volume of the rubberized material is 33. A cylinder has a surface area of 144π square inches about 304.1 cm3. and a height of 6 inches. What is the volume?

ANSWER: SOLUTION: 3 Use the surface area formula to solve for r. 304.1 cm Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

The radius is 6. Find the volume.

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume?

SOLUTION: Use the surface area formula to solve for r. ANSWER: 678.6 in3

The radius is 6. Find the volume.

Find the volume.

ANSWER: ANSWER: 3 3 678.6 in 576 in

inches. What is the volume? building. According to the specifications, the diameter SOLUTION: of the column can be between 30 centimeters and 95 Use the surface area formula to find the length of the centimeters. The height is to be 500 centimeters. base of the prism. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes. Find the volume.

ANSWER: 3 576 in ANSWER: 3,190,680.0 cm3 35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new 36. CCSS MODELING The base of a rectangular building. According to the specifications, the diameter swimming pool is sloped so one end of the pool is 6 of the column can be between 30 centimeters and 95 feet deep and the other end is 3 feet deep, as shown centimeters. The height is to be 500 centimeters. in the figure. If the width is 15 feet, find the volume What is the difference in volume between the largest of water it takes to fill the pool. and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively. SOLUTION:

The swimming pool is a combination of a rectangular

Find the difference between the volumes. prism and a trapezoidal prism. The base of the

rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms.

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown ANSWER: in the figure. If the width is 15 feet, find the volume 3 1575 ft of water it takes to fill the pool. 37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the same. What will be the height of the new containers? SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two SOLUTION:

prisms. Find the volume of the original container.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

ANSWER: 3 1575 ft

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each

drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Find the volume of the original container. SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

ANSWER:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius.

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

ANSWER: Sample answers:

ANSWER: 39. Find the volume of the regular pentagonal prism by Sample answers: dividing it into five equal triangular prisms. Describe

the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 12-4 Volumes of Prisms and Cylinders or 22 cm and a height of 10 cm.

39. Find the volume of the regular pentagonal prism by 40. PATIOS Mr. Thomas is planning to remove an old dividing it into five equal triangular prisms. Describe patio and install a new rectangular concrete patio 20 the base area and height of each triangular prism. feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards. SOLUTION: The base of the prism can be divided into 5 20 feet = yd congruent triangles of a base 8 cm and the 12 feet = 4 yd corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 4 in. = yd 10 cm. Find the base area of each triangular prism. Find the volume.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio Therefore, the second contractor is a less expensive

and $700 for removal of the old patio. Which is the option. less expensive option? Explain. ANSWER: SOLUTION: Because 2.96 yd3 of concrete are needed, the Convert all of the dimensions to yards. second contractor is less expensive at $2181.50.

MULTIPLE REPRESENTATIONS 20 feet = yd 41. In this problem, you will investigate right and oblique

12 feet = 4 yd cylinders 4 in. = yd . a. GEOMETRIC Draw a right cylinder and an Find the volume. oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects eSolutions Manual - Powered by Cognero the volume of the cylinder more: multiplying thePage 15 height by x or multiplying the radius by x. Explain. The total cost for the second contractor is about SOLUTION: . a. The oblique cylinder should look like the right cylinder (same height and size), except that it is Therefore, the second contractor is a less expensive pushed a little to the side, like a slinky. option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects

the volume of the cylinder more: multiplying the The volume of the square prism is greater. height by x or multiplying the radius by x. Explain. c. Do each scenario. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Find the volume of each.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less than x.

ANSWER: a.

The volume of the square prism is greater. c. Do each scenario.

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the

Assuming x > 1, multiplying the radius by x makes radius by x makes the volume x 2 times greater, 2 assuming x > 1. the volume x times greater.

CCSS CRITIQUE 2 42. Franciso and Valerie each For example, if x = 0.5, then x = 0.25, which is less calculated the volume of an equilateral triangular than x. prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your ANSWER: reasoning. a.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco b. Greater than; a square with a side length of 6 m used a different approach, but his solution is correct. 2 has an area of 36 m . A circle with a diameter of 6 Francisco used the standard formula for the volume m has an area of 9π or 28.3 m2. Since the heights of a solid, V = Bh. The area of the base, B, is one- are the same, the volume of the square prism is half the apothem multiplied by the perimeter of the greater. base.

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, ANSWER: assuming x > 1. Francisco; Valerie incorrectly used as the 42. CCSS CRITIQUE Franciso and Valerie each length of one side of the triangular base. Francisco calculated the volume of an equilateral triangular used a different approach, but his solution is correct. prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your 43. CHALLENGE A cylindrical can is used to fill a reasoning. container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

SOLUTION: b. square prism Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco c. triangular prism with a right triangle as the base used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base. SOLUTION:

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the ANSWER: container is 60π in 3. Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco a. Choose some basic values for 2 of the sides, and used a different approach, but his solution is correct. then determine the third side. Base: 3 by 5.

a. rectangular prism 3 by 5 by 4π b. square prism b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. c. triangular prism with a right triangle as the base

SOLUTION:

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the 5 by 5 by container is 60π in 3. c. Choose some basic values for 2 of the sides, and a. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. then determine the third side. Base: 3 by 5.

3 by 5 by 4π 3 by 4 by 10π

ANSWER: b. Choose some basic values for 2 of the sides, and Sample answers: then determine the third side. Base: 5 by 5.

a. 3 by 5 by 4π

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet 5 by 5 by gardening forum.

45. OPEN ENDED Draw and label a prism that has a b. 5 by 5 by volume of 50 cubic centimeters.

c. base with legs measuring 3 in. and 4 in., height SOLUTION: 10π in. Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest 44. WRITING IN MATH Write a helpful response to values to choose. the following question posted on an Internet gardening forum. Sample answer:

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order?

SOLUTION: ANSWER: Sample answer: The nursery means a cubic yard, 3 Sample answer: which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

SOLUTION: True; if two cylinders have the same height (h1 = h2) Choose 3 values that multiply to make 50. The and the same lateral area (L1 = L2), the circular factors of 50 are 2, 5, 5, so these are the simplest bases must have the same area.

values to choose.

Sample answer:

WRITING IN MATH 46. REASONING Determine whether the following 47. How are the formulas for statement is true or false. Explain. the volume of a prism and the volume of a cylinder

Two cylinders with the same height and the same similar? How are they different? lateral area must have the same volume. SOLUTION: SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, True; if two cylinders have the same height (h = h ) 1 2 so the expression representing the area varies, and the same lateral area (L1 = L2), the circular depending on the type of polygon it is. The base of a bases must have the same area. cylinder is a circle, so its area is πr2.

48. The volume of a triangular prism is 1380 cubic The radii must also be equal. centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is ANSWER: the height of the prism? True; if two cylinders have the same height and the A 34.5 cm same lateral area, the circular bases must have the B 2 23 cm same area. Therefore, πr h is the same for each C 17 cm cylinder. D 11.5 cm 47. WRITING IN MATH How are the formulas for SOLUTION: the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the ANSWER: area of the base by the height. The base of a prism is B a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 49. A cylindrical tank used for oil storage has a height 2 base of a cylinder is a circle, so its area is πr . that is half the length of its radius. If the volume of the tank is 1,122,360 ft3, what is the tank’s radius? 48. The volume of a triangular prism is 1380 cubic F 89.4 ft centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is G 178.8 ft the height of the prism? H 280.9 ft A 34.5 cm J 561.8 ft B 23 cm SOLUTION: C 17 cm D 11.5 cm SOLUTION:

ANSWER: ANSWER: B F

49. A cylindrical tank used for oil storage has a height 50. SHORT RESPONSE What is the ratio of the area that is half the length of its radius. If the volume of of the circle to the area of the square? the tank is 1,122,360 ft3, what is the tank’s radius? F 89.4 ft G 178.8 ft H 280.9 ft J 561.8 ft SOLUTION: SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

Therefore, the correct choice is C. SOLUTION: ANSWER: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can C

be written as shown. Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary. ANSWER:

52. 51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax SOLUTION: amount for a landowner whose property has a taxable value of $85,000? The lateral area L of a regular pyramid is , A $4.25 where is the slant height and P is the perimeter of B $170 the base. C $425 D $4250 The slant height is the height of each of the E $42,500 congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is ,

The perimeter is P = 3 × 10 or 30 feet.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

The perimeter is P = 3 × 10 or 30 feet.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 2 2 212.1 ft ; 255.4 ft So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

53. SOLUTION:

The lateral area L of a regular pyramid is , 2 So, the lateral area of the pyramid is about 212.1 ft . where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

Therefore, the surface area of the pyramid is about So, the lateral area of the pyramid is 126 cm2. 2 255.4 ft .

ANSWER: The surface area S of a regular 2 2 212.1 ft ; 255.4 ft pyramid is , where L is the lateral area and B is the area of the base.

53. SOLUTION: Therefore, the surface area of the pyramid is 175 The lateral area L of a regular pyramid is , cm2.

where is the slant height and P is the perimeter of ANSWER: the base. 2 2 Here, the base is a square of side 7 cm and the slant 126 cm ; 175 cm height is 9 cm.

54. SOLUTION: 2 So, the lateral area of the pyramid is 126 cm . The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the

The surface area S of a regular angle formed in the triangle below is 30°. pyramid is , where L is the lateral area and B is the area of the base.

Therefore, the surface area of the pyramid is 175 cm2. Use a trigonometric ratio to find the measure of the apothem a. ANSWER: 2 2 126 cm ; 175 cm

54. Find the lateral area and surface area of the pyramid. SOLUTION:

The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

So, the lateral area of the pyramid is 472.5 in2.

Use a trigonometric ratio to find the measure of the apothem a. Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

So, the lateral area of the pyramid is 472.5 in2. SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the indicated measure. Round to the Therefore, the surface area of the pyramid is about nearest tenth. 758.9 in2. 56. The area of a circle is 54 square meters. Find the

diameter. SOLUTION: ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

The diameter of the circle is about 8.3 m. SOLUTION: ANSWER: The area that needs to be coated is the sum of the 8.3 m lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 57. Find the diameter of a circle with an area of 102 2 9)(2) + 13(9) = 205 in . square centimeters. ANSWER: SOLUTION: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION: The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches.

SOLUTION:

The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

ANSWER: 7.8 ft 60. 59. Find the radius of a circle with an area of 271 square SOLUTION: inches. The area A of a kite is one half the product of the

SOLUTION: lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 ANSWER: 120 in 9.3 in. Find the area of each trapezoid, rhombus, or kite.

61. SOLUTION: The area A of a trapezoid is one half the product of 60. the height h and the sum of the lengths of its bases, SOLUTION: b1 and b2. The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 378 m

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

Find the volume of each prism.

3 The volume is 108 cm . ANSWER: 3 ANSWER: 26.95 m 108 cm3 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 3 396 in 5. SOLUTION: 3. the oblique rectangular prism shown at the right

SOLUTION: If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 3 the same volume. So, the volume of a right prism and 206.4 ft an oblique one of the same height and cross sectional area are same.

6. SOLUTION:

If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross 26.95 m3 sectional area are same. 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION:

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 5. 7.4 inches SOLUTION: SOLUTION:

ANSWER: ANSWER: 3 410.1 in 206.4 ft3 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? 6. A 4000 B 6400 SOLUTION: C 30,000 If two solids have the same height h and the same D 48,000 cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder SOLUTION: and an oblique one of the same height and cross sectional area are same.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. ANSWER: ANSWER: 3 1357.2 m D 7. a cylinder with a diameter of 16 centimeters and a CCSS SENSE-MAKING Find the volume of height of 5.1 centimeters each prism. SOLUTION:

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches

SOLUTION: ANSWER: 30 in3

ANSWER: 3 11. 410.1 in SOLUTION: 9. MULTIPLE CHOICE A rectangular lap pool The base is a triangle with a base length of 11 m and measures 80 feet long by 20 feet wide. If it needs to the corresponding height of 7 m. The height of the be filled to four feet deep and each cubic foot holds prism is 14 m. 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000

D 48,000 SOLUTION: ANSWER: 3 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000. 12.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

The height of the prism is 6 cm.

ANSWER: 30 in3 ANSWER: 324 cm3

11. SOLUTION: 13. The base is a triangle with a base length of 11 m and SOLUTION: the corresponding height of 7 m. The height of the If two solids have the same height h and the same prism is 14 m. cross-sectional area B at every level, then they have

the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. ANSWER: 3 2 539 m B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

ANSWER: 58.14 ft3

12. 14. an oblique hexagonal prism with a height of 15 SOLUTION: centimeters and with a base area of 136 square centimeters The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. SOLUTION: If two solids have the same height h and the same Use the Pythagorean Theorem to find the height of cross-sectional area B at every level, then they have the base. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER:

3 The height of the prism is 6 cm. 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches

13.

SOLUTION: ANSWER: If two solids have the same height h and the same 3 cross-sectional area B at every level, then they have 1534.25 in the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional CCSS SENSE-MAKING Find the volume of area are same. each cylinder. Round to the nearest tenth.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 16. 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is SOLUTION:

r = 5 yd and h = 18 yd

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters ANSWER: 3 SOLUTION: 1413.7 yd If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 2040 cm

Use the Pythagorean Theorem to find the height of the cylinder. ANSWER: 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16.

SOLUTION: r = 5 yd and h = 18 yd Now you can find the volume.

ANSWER: 3 1413.7 yd ANSWER: 3 823.0 in

17. SOLUTION: 19. r = 6 cm and h = 3.6 cm. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: r = 7.5 mm and h = 15.2 mm. 407.2 cm3

18. ANSWER: SOLUTION: 2686.1 mm3 r = 5.5 in.

20. PLANTER A planter is in the shape of a Use the Pythagorean Theorem to find the height of the cylinder. rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER:

3 2740.5 in ANSWER: 3 SHIPPING 823.0 in 21. A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box? 19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. SOLUTION: The volume of the empty space is the difference of r = 7.5 mm and h = 15.2 mm. volumes of the rectangular prism and the cylinders.

ANSWER: 3 2686.1 mm ANSWER: 3 20. PLANTER A planter is in the shape of a 521.5 cm

23.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm. ANSWER: 3 521.5 cm The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism. 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION: ANSWER: 3934.9 cm3 3 V = 10 ft and h = 2 ft Use the formula to find r.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 23. an oblique one of the same height and cross sectional SOLUTION: area are same. The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4

cm. Use the Pythagorean Theorem to find the height of the base. ANSWER: 3 48.9 m The height of the prism is 20 cm. 25. FOOD A cylindrical can of baked potato chips has a The volume V of a prism is V = Bh, where B is the height of 27 centimeters and a radius of 4 area of the base, h is the height of the prism. centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

ANSWER: 3934.9 cm3 SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

ANSWER: a. 3 When the height is tripled, h = 3h. 48.9 m

25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the

same radius, what is the height of the larger can?

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a When the height and the radius are tripled, the radius of 5 centimeters and a height of 8 centimeters. volume is multiplied by 27. Describe how each change affects the volume of the cylinder. d. When the dimensions are exchanged, r = 8 and h a. The height is tripled. = 5 cm. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

Compare to the original volume.

a. The volume is multiplied by . When the height is tripled, h = 3h.

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound So, when the radius is tripled, the volume is multiplied per square inch is desirable for root growth? Explain. by 9. c. If a bag of this soil holds 2.5 cubic feet, what is its

weight in pounds? c. When the height and the radius are tripled, r = 3r and h = 3h. SOLUTION:

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus: Compare to the original volume.

The volume is multiplied by . b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. ANSWER: a. The volume is multiplied by 3. c. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density ANSWER: 3 of the soil sample is the ratio of its weight to its a. 0.0019 lb / in volume. b. The plant should grow well in this soil since the a. If the weight of the container with the soil is 20 bulk density of 0.0019 lb / in3 is close to the desired pounds and the weight of the container alone is 5 3 pounds, what is the soil’s bulk density? b. Assuming bulk density of 0.0018 lb / in . that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound c. 8.3 lb per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its Find the volume of each composite solid. Round weight in pounds? to the nearest tenth if necessary. SOLUTION:

28. SOLUTION:

a. First calculate the volume of soil in the pot. Then The solid is a combination of two rectangular prisms. divide the weight of the soil by the volume. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs. ANSWER: 225 cm3 The soil density is thus:

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

c. 29. SOLUTION:

The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the 3 bulk density of 0.0019 lb / in is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb ANSWER: 3 Find the volume of each composite solid. Round 120 m to the nearest tenth if necessary.

ANSWER: ANSWER: 3 225 cm 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder? 29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm.

The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 ANSWER: volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder. 3 120 m

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the ANSWER: volume of the rubberized material that makes up the 5.7 cm holder? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches Find the volume.

and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

Find the difference between the volumes.

ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? ANSWER: 3,190,680.0 cm3 SOLUTION: Use the surface area formula to find the length of the 36. CCSS MODELING The base of a rectangular base of the prism. swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

Find the volume. SOLUTION:

The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two ANSWER: prisms. 3 576 in

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. ANSWER: 3 1575 ft SOLUTION: The volume will be the highest when the diameter is 37. CHANGING DIMENSIONS A soy milk company 95 cm and will be the lowest when it is 30 cm.That is is planning a promotion in which the volume of soy when the radii are 47.5 cm and 15 cm respectively. milk in each container will be increased by 25%. The company wants the base of the container to stay the Find the difference between the volumes. same. What will be the height of the new containers?

SOLUTION: Find the volume of the original container. ANSWER: 3,190,680.0 cm3

SOLUTION:

The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the ANSWER: rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two 38. DESIGN Sketch and label (in inches) three different prisms. designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

ANSWER: 3 1575 ft

SOLUTION:

Find the volume of the original container. If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

For any rectangular container, the volume equation is: ANSWER:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius. ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 ANSWER: 1100 cm ; Each triangular prism has a base area of Sample answers: 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

The total cost for the second contractor is about .

40. PATIOS Mr. Thomas is planning to remove an old c. ANALYTICAL Describe which change affects patio and install a new rectangular concrete patio 20 the volume of the cylinder more: multiplying the feet long, 12 feet wide, and 4 inches thick. One height by x or multiplying the radius by x. Explain. contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio SOLUTION: and $700 for removal of the old patio. Which is the a. The oblique cylinder should look like the right less expensive option? Explain. cylinder (same height and size), except that it is SOLUTION: pushed a little to the side, like a slinky. Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

b. Find the volume of each. Find the volume.

The total cost for the second contractor is about

. The volume of the square prism is greater.

c. Do each scenario. Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

12-4the Volumes cylinder? of Explain.Prisms and Cylinders ANSWER: c. ANALYTICAL Describe which change affects a. the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater. b. Find the volume of each. c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

CCSS CRITIQUE 42. Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning.

The volume of the square prism is greater. c. Do each scenario.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

Assuming x > 1, multiplying the radius by x makes

2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less ANSWER: than x. Francisco; Valerie incorrectly used as the ANSWER: length of one side of the triangular base. Francisco used a different approach, but his solution is correct. a. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the eSolutions Manual - Powered by Cognero Page 16 container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the SOLUTION: radius by x makes the volume x 2 times greater, 3 assuming x > 1. The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the CCSS CRITIQUE 42. Franciso and Valerie each 3 calculated the volume of an equilateral triangular container is 60π in . prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your a. Choose some basic values for 2 of the sides, and reasoning. then determine the third side. Base: 3 by 5.

SOLUTION:

Francisco; Valerie incorrectly used as the 3 by 5 by 4π length of one side of the triangular base. Francisco used a different approach, but his solution is correct. b. Choose some basic values for 2 of the sides, and Francisco used the standard formula for the volume then determine the third side. Base: 5 by 5. of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco 5 by 5 by used a different approach, but his solution is correct. c. Choose some basic values for 2 of the sides, and 43. CHALLENGE A cylindrical can is used to fill a then determine the third side. Base: Legs: 3 by 4. container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism

3 by 4 by 10π c. triangular prism with a right triangle as the base ANSWER: Sample answers:

a. 3 by 5 by 4π

SOLUTION: b. 5 by 5 by

3 The volume of the can is 20π in . It takes three full c. base with legs measuring 3 in. and 4 in., height cans to fill the container, so the volume of the 10π in. container is 60π in 3. WRITING IN MATH 44. Write a helpful response to the following question posted on an Internet a. Choose some basic values for 2 of the sides, and gardening forum. then determine the third side. Base: 3 by 5. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order?

SOLUTION: Sample answer: The nursery means a cubic yard, 3 3 by 5 by 4π which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine b. Choose some basic values for 2 of the sides, and the number of cubic yards of soil needed. then determine the third side. Base: 5 by 5. ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. SOLUTION: 5 by 5 by Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest values to choose. c. Choose some basic values for 2 of the sides, and

then determine the third side. Base: Legs: 3 by 4. Sample answer:

ANSWER:

3 by 4 by 10π Sample answer:

ANSWER: Sample answers:

a. 3 by 5 by 4π

bases must have the same area. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your

depending on the type of polygon it is. The base of a Sample answer: cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The ANSWER: 2 Sample answer: base of a cylinder is a circle, so its area is πr . 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm 46. REASONING Determine whether the following C 17 cm

statement is true or false. Explain. D 11.5 cm Two cylinders with the same height and the same lateral area must have the same volume. SOLUTION: SOLUTION:

True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: B

The radii must also be equal. 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of ANSWER: the tank is 1,122,360 ft3, what is the tank’s radius? True; if two cylinders have the same height and the F 89.4 ft same lateral area, the circular bases must have the G 178.8 ft 2 same area. Therefore, πr h is the same for each H 280.9 ft cylinder. J 561.8 ft 47. WRITING IN MATH How are the formulas for SOLUTION: the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 ANSWER: base of a cylinder is a circle, so its area is πr . F 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs 50. SHORT RESPONSE What is the ratio of the area measuring 8 centimeters and 15 centimeters. What is of the circle to the area of the square? the height of the prism? A 34.5 cm B 23 cm C 17 cm D 11.5 cm SOLUTION: The radius of the circle is 2x and the length of each SOLUTION: side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a ANSWER: taxable value of $85,000? B A $4.25 49. A cylindrical tank used for oil storage has a height B $170 that is half the length of its radius. If the volume of C $425 the tank is 1,122,360 ft3, what is the tank’s radius? D $4250 E F 89.4 ft $42,500 G 178.8 ft SOLUTION: H 280.9 ft Find the 0.5% of $85,000. J 561.8 ft SOLUTION: Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425

D $4250 Find the perimeter and area of the equilateral E $42,500 triangle for the base. Use the Pythagorean Theorem SOLUTION: to find the height h of the triangle. Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , The perimeter is P = 3 × 10 or 30 feet.

where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

2 So, the lateral area of the pyramid is about 212.1 ft .

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem Therefore, the surface area of the pyramid is about to find the height h of the triangle. 2 255.4 ft . ANSWER: 212.1 ft2; 255.4 ft2

53.

SOLUTION:

2 So, the lateral area of the pyramid is 126 cm . 2 So, the area of the base B is ft . The surface area S of a regular Find the lateral area L and surface area S of the pyramid is , where regular pyramid. L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is about 212.1 ft2. Therefore, the surface area of the pyramid is 175

cm2.

ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 2 255.4 ft . 54.

53. SOLUTION:

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 54. 472.5 in2; 758.9 in2 SOLUTION: The pyramid has a slant height of 15 inches and the 55. BAKING Many baking pans are given a special base is a hexagon with sides of 10.5 inches. nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of A central angle of the hexagon is or 60°, so the the inside of the pan that needs to be coated? angle formed in the triangle below is 30°.

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

Use a trigonometric ratio to find the measure of the ANSWER: apothem a. 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter.

SOLUTION: Find the lateral area and surface area of the pyramid.

2 So, the lateral area of the pyramid is 472.5 in . The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters.

SOLUTION: Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches

by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated? The diameter of the circle is about 11.4 m. ANSWER: 11.4 cm

58. The area of a circle is 191 square feet. Find the radius. SOLUTION: The area that needs to be coated is the sum of the SOLUTION: lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

The diameter of the circle is about 8.3 m.

ANSWER:

8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. ANSWER: 9.3 in. SOLUTION: Find the area of each trapezoid, rhombus, or kite.

60.

SOLUTION: The diameter of the circle is about 11.4 m. The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2. ANSWER: 11.4 cm d1 = 12 in. and d2 = 7 + 13 = 20 in. 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 2 120 in

ANSWER: 7.8 ft 61. 59. Find the radius of a circle with an area of 271 square SOLUTION: inches. The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, SOLUTION: b1 and b2.

ANSWER: 2 378 m ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

Find the volume of each prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 3 396 in 5. SOLUTION: 3. the oblique rectangular prism shown at the right

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right cylinder 3 and an oblique one of the same height and cross 26.95 m sectional area are same. 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 3 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION:

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 5. 7.4 inches SOLUTION: SOLUTION:

ANSWER: ANSWER: 3 410.1 in 206.4 ft3 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? 6. A 4000 B 6400 SOLUTION: C 30,000 If two solids have the same height h and the same D 48,000 cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder SOLUTION: and an oblique one of the same height and cross sectional area are same.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. ANSWER: ANSWER: 3 1357.2 m D 7. a cylinder with a diameter of 16 centimeters and a CCSS SENSE-MAKING Find the volume of

height of 5.1 centimeters each prism. SOLUTION:

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches

SOLUTION: ANSWER: 30 in3

ANSWER: 3 410.1 in 11. SOLUTION: MULTIPLE CHOICE 9. A rectangular lap pool The base is a triangle with a base length of 11 m and measures 80 feet long by 20 feet wide. If it needs to the corresponding height of 7 m. The height of the be filled to four feet deep and each cubic foot holds prism is 14 m. 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000

D 48,000 SOLUTION: ANSWER: 3 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000. 12.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

The height of the prism is 6 cm.

ANSWER: 30 in3 ANSWER: 324 cm3

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. ANSWER: 3 2 539 m B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

ANSWER: 58.14 ft3

ANSWER:

3 The height of the prism is 6 cm. 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches

13.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 16. 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is SOLUTION:

r = 5 yd and h = 18 yd

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square

centimeters ANSWER: 3 SOLUTION: 1413.7 yd If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 2040 cm

Use the Pythagorean Theorem to find the height of the cylinder. ANSWER: 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. SOLUTION:

r = 5 yd and h = 18 yd Now you can find the volume.

ANSWER: 3 1413.7 yd ANSWER: 3 823.0 in

17. SOLUTION:

r = 6 cm and h = 3.6 cm. 19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: r = 7.5 mm and h = 15.2 mm. 407.2 cm3

18. ANSWER: SOLUTION: 3 r = 5.5 in. 2686.1 mm

20. PLANTER A planter is in the shape of a Use the Pythagorean Theorem to find the height of the cylinder. rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER: 2686.1 mm3 ANSWER: 20. PLANTER A planter is in the shape of a 521.5 cm3

23.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm. ANSWER:

Therefore, the diameter of the cylinders should be 24.

about 2.52 ft. SOLUTION: ANSWER: The circular bases at the top and bottom of the net 2.52 ft indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. Find the volume of the solid formed by each However, since the middle piece is a parallelogram, it net. is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height

of the base. ANSWER: 3 48.9 m The height of the prism is 20 cm. 25. FOOD A cylindrical can of baked potato chips has a The volume V of a prism is V = Bh, where B is the height of 27 centimeters and a radius of 4 area of the base, h is the height of the prism. centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

ANSWER: 3934.9 cm3 SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

ANSWER: a. 3 When the height is tripled, h = 3h. 48.9 m

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a When the height and the radius are tripled, the radius of 5 centimeters and a height of 8 centimeters. volume is multiplied by 27. Describe how each change affects the volume of the cylinder. d. When the dimensions are exchanged, r = 8 and h a. The height is tripled. = 5 cm. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

Compare to the original volume.

a. The volume is multiplied by . When the height is tripled, h = 3h.

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound So, when the radius is tripled, the volume is multiplied per square inch is desirable for root growth? Explain. by 9. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? c. When the height and the radius are tripled, r = 3r and h = 3h. SOLUTION:

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus: Compare to the original volume.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density ANSWER: 3 of the soil sample is the ratio of its weight to its a. 0.0019 lb / in volume. b. The plant should grow well in this soil since the a. If the weight of the container with the soil is 20 3 pounds and the weight of the container alone is 5 bulk density of 0.0019 lb / in is close to the desired 3 pounds, what is the soil’s bulk density? b. Assuming bulk density of 0.0018 lb / in . that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound c. 8.3 lb per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its Find the volume of each composite solid. Round weight in pounds? to the nearest tenth if necessary. SOLUTION:

28. SOLUTION:

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs. ANSWER: 225 cm3 The soil density is thus:

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

c . 29. SOLUTION:

The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the 3 bulk density of 0.0019 lb / in is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb ANSWER: 3 Find the volume of each composite solid. Round 120 m to the nearest tenth if necessary.

ANSWER: ANSWER: 3 225 cm 3 260.5 in

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder? 29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm.

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the ANSWER: volume of the rubberized material that makes up the 5.7 cm holder? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches Find the volume. and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

tenth cubic centimeter. SOLUTION: The radius is 6. Find the volume. The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 678.6 in3

Find the volume. SOLUTION:

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. ANSWER: 3 SOLUTION: 1575 ft The volume will be the highest when the diameter is 37. CHANGING DIMENSIONS A soy milk company 95 cm and will be the lowest when it is 30 cm.That is is planning a promotion in which the volume of soy when the radii are 47.5 cm and 15 cm respectively. milk in each container will be increased by 25%. The company wants the base of the container to stay the Find the difference between the volumes. same. What will be the height of the new containers?

SOLUTION: Find the volume of the original container. ANSWER: 3,190,680.0 cm3

For any cylindrical container, we have the following equation for volume:

ANSWER: 3 1575 ft

SOLUTION:

Find the volume of the original container. If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

For any rectangular container, the volume equation is: ANSWER:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius. ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 ANSWER: 1100 cm ; Each triangular prism has a base area of Sample answers: 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

The total cost for the second contractor is about .

40. PATIOS Mr. Thomas is planning to remove an old c. ANALYTICAL patio and install a new rectangular concrete patio 20 Describe which change affects feet long, 12 feet wide, and 4 inches thick. One the volume of the cylinder more: multiplying the

contractor bid $2225 for the project. A second height by x or multiplying the radius by x. Explain. contractor bid $500 per cubic yard for the new patio SOLUTION: and $700 for removal of the old patio. Which is the a. The oblique cylinder should look like the right

less expensive option? Explain. cylinder (same height and size), except that it is SOLUTION: pushed a little to the side, like a slinky. Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

b. Find the volume of each. Find the volume.

The total cost for the second contractor is about . The volume of the square prism is greater.

c. Do each scenario. Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders . a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters. Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater. b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume For example, if x = 0.5, then x2 = 0.25, which is less greater than, less than, or equal to the volume of than x. the cylinder? Explain. ANSWER: c. ANALYTICAL Describe which change affects a. the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater. b. Find the volume of each. c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

The volume of the square prism is greater. c. Do each scenario.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

For example, if x = 0.5, then x2 = 0.25, which is less ANSWER: than x. Francisco; Valerie incorrectly used as the ANSWER: length of one side of the triangular base. Francisco used a different approach, but his solution is correct. a. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the SOLUTION: radius by x makes the volume x 2 times greater, assuming x > 1. 3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the 42. CCSS CRITIQUE Franciso and Valerie each 3 calculated the volume of an equilateral triangular container is 60π in . prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your a. Choose some basic values for 2 of the sides, and reasoning. then determine the third side. Base: 3 by 5.

SOLUTION: Francisco; Valerie incorrectly used as the

length of one side of the triangular base. Francisco 3 by 5 by 4π used a different approach, but his solution is correct. b. Choose some basic values for 2 of the sides, and Francisco used the standard formula for the volume then determine the third side. Base: 5 by 5. of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco 5 by 5 by 12-4used Volumes a different of Prisms approach, and butCylinders his solution is correct. c. Choose some basic values for 2 of the sides, and 43. CHALLENGE A cylindrical can is used to fill a then determine the third side. Base: Legs: 3 by 4. container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism

3 by 4 by 10π c. triangular prism with a right triangle as the base ANSWER: Sample answers:

a. 3 by 5 by 4π

SOLUTION: b. 5 by 5 by

3 The volume of the can is 20π in . It takes three full c. base with legs measuring 3 in. and 4 in., height cans to fill the container, so the volume of the 10π in. container is 60π in 3. 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet a. Choose some basic values for 2 of the sides, and gardening forum.

then determine the third side. Base: 3 by 5. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 3 by 5 by 4π which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine b. Choose some basic values for 2 of the sides, and the number of cubic yards of soil needed. then determine the third side. Base: 5 by 5. ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

then determine the third side. Base: Legs: 3 by 4. Sample answer: eSolutions Manual - Powered by Cognero Page 17

ANSWER:

3 by 4 by 10π Sample answer:

ANSWER: Sample answers:

a. 3 by 5 by 4π

44. WRITING IN MATH Write a helpful response to True; if two cylinders have the same height (h1 = h2) the following question posted on an Internet and the same lateral area (L = L ), the circular gardening forum. 1 2 bases must have the same area. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine The radii must also be equal. the number of cubic yards of soil needed. ANSWER: ANSWER: True; if two cylinders have the same height and the Sample answer: The nursery means a cubic yard, 3 same lateral area, the circular bases must have the which is 3 or 27 cubic feet. Find the volume of your 2 same area. Therefore, πr h is the same for each garden in cubic feet and divide by 27 to determine cylinder. the number of cubic yards of soil needed. 47. WRITING IN MATH How are the formulas for 45. OPEN ENDED Draw and label a prism that has a the volume of a prism and the volume of a cylinder volume of 50 cubic centimeters. similar? How are they different? SOLUTION: SOLUTION: Choose 3 values that multiply to make 50. The Both formulas involve multiplying the area of the factors of 50 are 2, 5, 5, so these are the simplest base by the height. The base of a prism is a polygon, values to choose. so the expression representing the area varies,

depending on the type of polygon it is. The base of a Sample answer: cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The ANSWER: 2 Sample answer: base of a cylinder is a circle, so its area is πr . 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm 46. REASONING Determine whether the following C 17 cm statement is true or false. Explain. D 11.5 cm Two cylinders with the same height and the same lateral area must have the same volume. SOLUTION: SOLUTION:

True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: B

The radii must also be equal. 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of ANSWER: the tank is 1,122,360 ft3, what is the tank’s radius? True; if two cylinders have the same height and the F 89.4 ft same lateral area, the circular bases must have the G 178.8 ft 2 same area. Therefore, πr h is the same for each H 280.9 ft cylinder. J 561.8 ft 47. WRITING IN MATH How are the formulas for SOLUTION: the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 ANSWER: base of a cylinder is a circle, so its area is πr . F 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs 50. SHORT RESPONSE What is the ratio of the area measuring 8 centimeters and 15 centimeters. What is of the circle to the area of the square? the height of the prism? A 34.5 cm B 23 cm C 17 cm D 11.5 cm SOLUTION: The radius of the circle is 2x and the length of each SOLUTION: side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a ANSWER: taxable value of $85,000? B A $4.25 49. A cylindrical tank used for oil storage has a height B $170 that is half the length of its radius. If the volume of C $425 the tank is 1,122,360 ft3, what is the tank’s radius? D $4250 F 89.4 ft E $42,500 G 178.8 ft SOLUTION: H 280.9 ft Find the 0.5% of $85,000. J 561.8 ft SOLUTION: Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425

D $4250 Find the perimeter and area of the equilateral E $42,500 triangle for the base. Use the Pythagorean Theorem SOLUTION: to find the height h of the triangle. Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , The perimeter is P = 3 × 10 or 30 feet.

where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

2 So, the lateral area of the pyramid is about 212.1 ft .

Find the perimeter and area of the equilateral

triangle for the base. Use the Pythagorean Theorem Therefore, the surface area of the pyramid is about to find the height h of the triangle. 2 255.4 ft . ANSWER: 212.1 ft2; 255.4 ft2

53.

SOLUTION:

2 So, the lateral area of the pyramid is 126 cm .

So, the area of the base B is ft2.

The surface area S of a regular Find the lateral area L and surface area S of the pyramid is , where regular pyramid.

L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is about 212.1 ft2. Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 2 255.4 ft . 54.

53. SOLUTION:

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 54. 472.5 in2; 758.9 in2 SOLUTION: The pyramid has a slant height of 15 inches and the 55. BAKING Many baking pans are given a special base is a hexagon with sides of 10.5 inches. nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of A central angle of the hexagon is or 60°, so the the inside of the pan that needs to be coated? angle formed in the triangle below is 30°.

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

Use a trigonometric ratio to find the measure of the ANSWER: apothem a. 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter.

SOLUTION: Find the lateral area and surface area of the pyramid.

2 So, the lateral area of the pyramid is 472.5 in . The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters.

SOLUTION: Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated? The diameter of the circle is about 11.4 m. ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION: The area that needs to be coated is the sum of the SOLUTION: lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. ANSWER: 9.3 in. SOLUTION: Find the area of each trapezoid, rhombus, or kite.

60.

ANSWER: 2 120 in

ANSWER: 2 378 m ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

Find the volume of each prism.

SOLUTION: 1. If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have The volume V of a prism is V = Bh, where B is the the same volume. So, the volume of a right prism and area of a base and h is the height of the prism. an oblique one of the same height and cross sectional area are same.

3 The volume is 108 cm .

ANSWER: ANSWER: 26.95 m3 108 cm3 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 2. area are same.

SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the

nearest tenth.

ANSWER: 3 396 in 5. 3. the oblique rectangular prism shown at the right SOLUTION:

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters

SOLUTION:

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 5. 7.4 inches SOLUTION: SOLUTION:

ANSWER: ANSWER: 3 3 410.1 in 206.4 ft 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? 6. A 4000 SOLUTION: B 6400 If two solids have the same height h and the same C 30,000 cross-sectional area B at every level, then they have D 48,000 the same volume. So, the volume of a right cylinder SOLUTION: and an oblique one of the same height and cross sectional area are same.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. ANSWER: 3 1357.2 m ANSWER: D 7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters CCSS SENSE-MAKING Find the volume of each prism. SOLUTION:

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches

SOLUTION: ANSWER: 30 in3

ANSWER: 3 410.1 in 11.

9. MULTIPLE CHOICE A rectangular lap pool SOLUTION: measures 80 feet long by 20 feet wide. If it needs to The base is a triangle with a base length of 11 m and be filled to four feet deep and each cubic foot holds the corresponding height of 7 m. The height of the 7.5 gallons, how many gallons will it take to fill the prism is 14 m. lap pool? A 4000 B 6400 C 30,000 D 48,000

SOLUTION: ANSWER: 3 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000. 12. Therefore, the correct choice is D. SOLUTION: ANSWER: The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. D

CCSS SENSE-MAKING Find the volume of Use the Pythagorean Theorem to find the height of each prism. the base.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

The height of the prism is 6 cm.

ANSWER: 30 in3 ANSWER: 324 cm3

11. SOLUTION: 13. The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the SOLUTION: prism is 14 m. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the ANSWER: area of a base and h is the height of the prism. 3 539 m 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

ANSWER: 58.14 ft3 12. 14. an oblique hexagonal prism with a height of 15 SOLUTION: centimeters and with a base area of 136 square The base is a right triangle with a leg length of 9 cm centimeters and the hypotenuse of length 15 cm. SOLUTION: Use the Pythagorean Theorem to find the height of If two solids have the same height h and the same the base. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 The height of the prism is 6 cm. 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: ANSWER: If two solids have the same height h and the same 3 cross-sectional area B at every level, then they have 324 cm the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

13.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 16. B = 11.4 ft and h = 5.1 ft. Therefore, the volume is SOLUTION: r = 5 yd and h = 18 yd ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters ANSWER: SOLUTION: 3 If two solids have the same height h and the same 1413.7 yd cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 2040 cm

18. SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of ANSWER: the cylinder.

1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. SOLUTION: r = 5 yd and h = 18 yd

Now you can find the volume.

ANSWER: 3 1413.7 yd ANSWER: 3 823.0 in

17. SOLUTION: r = 6 cm and h = 3.6 cm. 19.

18. SOLUTION: ANSWER: r = 5.5 in. 2686.1 mm3

Use the Pythagorean Theorem to find the height of 20. PLANTER A planter is in the shape of a the cylinder. rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER: 3 ANSWER: 2740.5 in 3 823.0 in 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box? 19. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. SOLUTION: r = 7.5 mm and h = 15.2 mm. The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

ANSWER:

2686.1 mm3 ANSWER: 20. PLANTER A planter is in the shape of a 3 521.5 cm rectangular prism 18 inches long, inches deep, 22. SANDCASTLES In a sandcastle competition, and 12 inches high. What is the volume of potting soil contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct in the planter if the planter is filled to inches amount of sand, they want to create cylinders that below the top? are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION: SOLUTION: The planter is to be filled inches below the top, so 3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: Therefore, the diameter of the cylinders should be 3 2740.5 in about 2.52 ft.

21. SHIPPING A box 18 centimeters by 9 centimeters ANSWER: by 15 centimeters is being used to ship two 2.52 ft cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown Find the volume of the solid formed by each at the right. What is the volume of the empty space net. in the box?

23. SOLUTION: SOLUTION: The volume of the empty space is the difference of The middle piece of the net is the front of the solid. volumes of the rectangular prism and the cylinders. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

ANSWER: The height of the prism is 20 cm. 3 521.5 cm The volume V of a prism is V = Bh, where B is the 22. SANDCASTLES In a sandcastle competition, area of the base, h is the height of the prism. contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant.

What should the diameter of the cylinders be? SOLUTION: ANSWER: 3 3 3934.9 cm V = 10 ft and h = 2 ft Use the formula to find r.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

ANSWER: 3 3934.9 cm SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

24. SOLUTION: The circular bases at the top and bottom of the net

indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. The height of the new can will be 35.1 cm. However, since the middle piece is a parallelogram, it is oblique. ANSWER:

35.1 cm The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m. 26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. If two solids have the same height h and the same Describe how each change affects the volume of the cross-sectional area B at every level, then they have cylinder. the same volume. So, the volume of a right prism and a. The height is tripled. an oblique one of the same height and cross sectional b. The radius is tripled. area are same. c. Both the radius and the height are tripled.

d. The dimensions are exchanged. SOLUTION:

ANSWER: a. 3 48.9 m When the height is tripled, h = 3h.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. When the height and the radius are tripled, the Describe how each change affects the volume of the volume is multiplied by 27. cylinder. a. The height is tripled. d. When the dimensions are exchanged, r = 8 and h b. The radius is tripled. = 5 cm. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

Compare to the original volume.

a. When the height is tripled, h = 3h. The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a So, when the radius is tripled, the volume is multiplied plant grow in this soil if a bulk density of 0.018 pound by 9. per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its c. When the height and the radius are tripled, r = 3r weight in pounds?

and h = 3h. SOLUTION:

a. First calculate the volume of soil in the pot. Then When the height and the radius are tripled, the divide the weight of the soil by the volume. volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus: Compare to the original volume.

The volume is multiplied by . b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant ANSWER: should grow fairly well. a. The volume is multiplied by 3. c. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density ANSWER: of the soil sample is the ratio of its weight to its 3 volume. a. 0.0019 lb / in

a. If the weight of the container with the soil is 20 b. The plant should grow well in this soil since the pounds and the weight of the container alone is 5 bulk density of 0.0019 lb / in3 is close to the desired 3 pounds, what is the soil’s bulk density? b. Assuming bulk density of 0.0018 lb / in . that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound c. 8.3 lb per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its Find the volume of each composite solid. Round weight in pounds? to the nearest tenth if necessary. SOLUTION:

28.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs. ANSWER: 3 The soil density is thus: 225 cm

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

c. 29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb ANSWER: Find the volume of each composite solid. Round 3 to the nearest tenth if necessary. 120 m

28. 30. SOLUTION: SOLUTION: The solid is a combination of two rectangular prisms. The solid is a combination of a rectangular prism and The base of one rectangular prism is 5 cm by 3 cm two half cylinders. and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: ANSWER: 225 cm3 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder? 29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a

radius of and a height of 11.5 cm.

The empty space used to keep the can has a radius

of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 ANSWER: volume V of a cylinder is V = πr h, where r is the 3 120 m radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is 30. about 304.1 cm3.

SOLUTION: ANSWER: The solid is a combination of a rectangular prism and 3 two half cylinders. 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the ANSWER: volume of the rubberized material that makes up the 5.7 cm holder? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth.

32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 ANSWER: centimeters. What is the height? 678.6 in3 SOLUTION: 34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches Find the volume. and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

The radius is 6. Find the volume. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? ANSWER: SOLUTION: 3,190,680.0 cm3 Use the surface area formula to find the length of the base of the prism. 36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. ANSWER: 3 SOLUTION: 1575 ft The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is 37. CHANGING DIMENSIONS A soy milk company when the radii are 47.5 cm and 15 cm respectively. is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The Find the difference between the volumes. company wants the base of the container to stay the same. What will be the height of the new containers?

SOLUTION: ANSWER: Find the volume of the original container.

3,190,680.0 cm3

For any cylindrical container, we have the following equation for volume:

ANSWER: 3 1575 ft

37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy

milk in each container will be increased by 25%. The company wants the base of the container to stay the The last equation gives us a relation between the same. What will be the height of the new containers? radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height.

SOLUTION: Find the volume of the original container.

If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

For any rectangular container, the volume equation is: ANSWER:

Choose numbers for any two of the dimensions and 38. DESIGN Sketch and label (in inches) three different we can solve for the third. Let l = 2.25 in. and w = designs for a dry ingredient measuring cup that holds 2.5 in. 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius. ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: ANSWER: 3 1100 cm ; Each triangular prism has a base area of Sample answers: 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option. SOLUTION: ANSWER: The base of the prism can be divided into 5 3 congruent triangles of a base 8 cm and the Because 2.96 yd of concrete are needed, the corresponding height 5.5 cm. So, the pentagonal second contractor is less expensive at $2181.50. prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism. 41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders . Therefore, the volume of the pentagonal prism is a. GEOMETRIC Draw a right cylinder and an

oblique cylinder with a height of 10 meters and a diameter of 6 meters. ANSWER: 3 1100 cm ; Each triangular prism has a base area of b. VERBAL A square prism has a height of 10 2 meters and a base edge of 6 meters. Is its volume or 22 cm and a height of 10 cm. greater than, less than, or equal to the volume of the cylinder? Explain. 40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 c. ANALYTICAL Describe which change affects feet long, 12 feet wide, and 4 inches thick. One the volume of the cylinder more: multiplying the contractor bid $2225 for the project. A second height by x or multiplying the radius by x. Explain. contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the SOLUTION: less expensive option? Explain. a. The oblique cylinder should look like the right cylinder (same height and size), except that it is SOLUTION: pushed a little to the side, like a slinky. Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume. b. Find the volume of each.

The total cost for the second contractor is about . The volume of the square prism is greater. Therefore, the second contractor is a less expensive c. Do each scenario. option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders . a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a

diameter of 6 meters. Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater. b. VERBAL A square prism has a height of 10

meters and a base edge of 6 meters. Is its volume 2 greater than, less than, or equal to the volume of For example, if x = 0.5, then x = 0.25, which is less the cylinder? Explain. than x.

ANSWER: c. ANALYTICAL Describe which change affects a. the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

The volume of the square prism is greater. c. Do each scenario.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater.

a. rectangular prism

b. Greater than; a square with a side length of 6 m b. square prism 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights c. triangular prism with a right triangle as the base are the same, the volume of the square prism is greater.

assuming x > 1. 3 The volume of the can is 20π in . It takes three full 42. CCSS CRITIQUE Franciso and Valerie each cans to fill the container, so the volume of the calculated the volume of an equilateral triangular container is 60π in 3. prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your a. Choose some basic values for 2 of the sides, and reasoning. then determine the third side. Base: 3 by 5.

SOLUTION:

Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco 3 by 5 by 4π used a different approach, but his solution is correct. b. Choose some basic values for 2 of the sides, and Francisco used the standard formula for the volume then determine the third side. Base: 5 by 5. of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

ANSWER:

a. rectangular prism

b. square prism

c. triangular prism with a right triangle as the base 3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π

SOLUTION: b. 5 by 5 by 3 The volume of the can is 20π in . It takes three full c. base with legs measuring 3 in. and 4 in., height cans to fill the container, so the volume of the 10π in. container is 60π in 3. 44. WRITING IN MATH Write a helpful response to a. Choose some basic values for 2 of the sides, and the following question posted on an Internet then determine the third side. Base: 3 by 5. gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION:

Sample answer: The nursery means a cubic yard, 3 by 5 by 4π 3 which is 3 or 27 cubic feet. Find the volume of your b. Choose some basic values for 2 of the sides, and garden in cubic feet and divide by 27 to determine then determine the third side. Base: 5 by 5. the number of cubic yards of soil needed. ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters.

ANSWER: 3 by 4 by 10π Sample answer: ANSWER: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by 46. REASONING Determine whether the following statement is true or false. Explain. Two cylinders with the same height and the same c. base with legs measuring 3 in. and 4 in., height lateral area must have the same volume. 10π in. SOLUTION: 44. WRITING IN MATH Write a helpful response to True; if two cylinders have the same height (h = h ) the following question posted on an Internet 1 2 gardening forum. and the same lateral area (L1 = L2), the circular bases must have the same area. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. The radii must also be equal.

ANSWER: ANSWER: Sample answer: The nursery means a cubic yard, True; if two cylinders have the same height and the which is 33 or 27 cubic feet. Find the volume of your same lateral area, the circular bases must have the 2 12-4garden Volumes in cubic of Prisms feet and and divide Cylinders by 27 to determine same area. Therefore, πr h is the same for each the number of cubic yards of soil needed. cylinder.

48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm 46. REASONING Determine whether the following B 23 cm statement is true or false. Explain. C 17 cm Two cylinders with the same height and the same D 11.5 cm lateral area must have the same volume. SOLUTION: SOLUTION: True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: B

The radii must also be equal. 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of ANSWER: 3 True; if two cylinders have the same height and the the tank is 1,122,360 ft , what is the tank’s radius? same lateral area, the circular bases must have the F 89.4 ft 2 G 178.8 ft same area. Therefore, πr h is the same for each H cylinder. 280.9 ft J 561.8 ft 47. WRITING IN MATH How are the formulas for SOLUTION: the volume of a prism and the volume of a cylinder eSolutionssimilar?Manual How- Powered are theyby Cognero different? Page 18 SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr . ANSWER: F 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs 50. SHORT RESPONSE What is the ratio of the area measuring 8 centimeters and 15 centimeters. What is of the circle to the area of the square? the height of the prism? A 34.5 cm B 23 cm C 17 cm D 11.5 cm SOLUTION: SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax ANSWER: amount for a landowner whose property has a B taxable value of $85,000? A $4.25 49. A cylindrical tank used for oil storage has a height B $170 that is half the length of its radius. If the volume of C $425 3 the tank is 1,122,360 ft , what is the tank’s radius? D $4250 F 89.4 ft E $42,500 G 178.8 ft SOLUTION: H 280.9 ft Find the 0.5% of $85,000. J 561.8 ft

SOLUTION: Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52.

SOLUTION: ANSWER: The lateral area L of a regular pyramid is , F where is the slant height and P is the perimeter of 50. SHORT RESPONSE What is the ratio of the area the base. of the circle to the area of the square? The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 D $4250 E $42,500 Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem SOLUTION: to find the height h of the triangle. Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , The perimeter is P = 3 × 10 or 30 feet. where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

ANSWER: 212.1 ft2; 255.4 ft2

53. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of

the base. The perimeter is P = 3 × 10 or 30 feet. Here, the base is a square of side 7 cm and the slant

height is 9 cm.

So, the lateral area of the pyramid is 126 cm2. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the The surface area S of a regular regular pyramid. pyramid is , where L is the lateral area and B is the area of the base.

2 So, the lateral area of the pyramid is about 212.1 ft . Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

53. SOLUTION:

2 So, the lateral area of the pyramid is 126 cm . Find the lateral area and surface area of the pyramid. The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm Therefore, the surface area of the pyramid is about 758.9 in2.

54. ANSWER: 472.5 in2; 758.9 in2 SOLUTION: The pyramid has a slant height of 15 inches and the 55. BAKING Many baking pans are given a special base is a hexagon with sides of 10.5 inches. nonstick coating. A rectangular cake pan is 9 inches A central angle of the hexagon is or 60°, so the by 13 inches by 2 inches deep. What is the area of angle formed in the triangle below is 30°. the inside of the pan that needs to be coated?

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in . Use a trigonometric ratio to find the measure of the ANSWER: apothem a. 2 205 in Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter.

SOLUTION: Find the lateral area and surface area of the pyramid.

So, the lateral area of the pyramid is 472.5 in2.

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters.

Therefore, the surface area of the pyramid is about SOLUTION: 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the SOLUTION: radius. The area that needs to be coated is the sum of the SOLUTION: lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102

square centimeters. ANSWER: SOLUTION: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60.

SOLUTION: The diameter of the circle is about 11.4 m. The area A of a kite is one half the product of the lengths of its diagonals, d and d . ANSWER: 1 2 11.4 cm d1 = 12 in. and d2 = 7 + 13 = 20 in. 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 2 120 in

ANSWER:

2 378 m ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: Find the volume of each prism. 3 396 in

3. the oblique rectangular prism shown at the right

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

3 The volume is 108 cm .

ANSWER: 108 cm3

ANSWER: 26.95 m3

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters

2. SOLUTION: If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have The volume V of a prism is V = Bh, where B is the the same volume. So, the volume of a right prism and area of a base and h is the height of the prism. an oblique one of the same height and cross sectional area are same.

ANSWER: 3 218.4 cm ANSWER: Find the volume of each cylinder. Round to the 3 396 in nearest tenth. 3. the oblique rectangular prism shown at the right

5. SOLUTION: SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 206.4 ft3

ANSWER: 6. 26.95 m3 SOLUTION: 4. an oblique pentagonal prism with a base area of 42 If two solids have the same height h and the same square centimeters and a height of 5.2 centimeters cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross If two solids have the same height h and the same sectional area are same. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 1357.2 m ANSWER: 7. a cylinder with a diameter of 16 centimeters and a 3 218.4 cm height of 5.1 centimeters

Find the volume of each cylinder. Round to the SOLUTION: nearest tenth.

5. SOLUTION: ANSWER: 1025.4 cm3 8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION:

ANSWER: 206.4 ft3

ANSWER: 3 410.1 in

6. 9. MULTIPLE CHOICE A rectangular lap pool SOLUTION: measures 80 feet long by 20 feet wide. If it needs to If two solids have the same height h and the same be filled to four feet deep and each cubic foot holds cross-sectional area B at every level, then they have 7.5 gallons, how many gallons will it take to fill the

the same volume. So, the volume of a right cylinder lap pool? and an oblique one of the same height and cross A 4000 sectional area are same. B 6400 C 30,000 D 48,000 SOLUTION:

ANSWER: Each cubic foot holds 7.5 gallons of water. So, the 3 1357.2 m amount of water required to fill the pool is 6400(7.5) = 48,000. 7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters Therefore, the correct choice is D.

SOLUTION: ANSWER: D CCSS SENSE-MAKING Find the volume of each prism.

ANSWER: 10. 1025.4 cm3 SOLUTION: 8. a cylinder with a radius of 4.2 inches and a height of The base is a rectangle of length 3 in. and width 2 in. 7.4 inches The height of the prism is 5 in. SOLUTION:

ANSWER: 3 ANSWER: 30 in 3 410.1 in

9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the 11. lap pool? A 4000 SOLUTION: B 6400 The base is a triangle with a base length of 11 m and C 30,000 the corresponding height of 7 m. The height of the prism is 14 m. D 48,000

SOLUTION:

Each cubic foot holds 7.5 gallons of water. So, the ANSWER: 3 amount of water required to fill the pool is 6400(7.5) 539 m = 48,000.

Therefore, the correct choice is D.

ANSWER: D 12. CCSS SENSE-MAKING Find the volume of SOLUTION: each prism. The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm.

10. Use the Pythagorean Theorem to find the height of the base. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

ANSWER: 3 The height of the prism is 6 cm. 30 in

ANSWER: 11. 324 cm3 SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the prism is 14 m.

13. SOLUTION:

If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and 3 an oblique one of the same height and cross sectional 539 m area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

12. SOLUTION: The base is a right triangle with a leg length of 9 cm ANSWER: and the hypotenuse of length 15 cm. 58.14 ft3

Use the Pythagorean Theorem to find the height of 14. an oblique hexagonal prism with a height of 15 the base. centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The height of the prism is 6 cm.

ANSWER: 3 2040 cm ANSWER: 3 15. a square prism with a base edge of 9.5 inches and a 324 cm height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 13. area are same. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: The volume V of a prism is V = Bh, where B is the 1534.25 in3 area of a base and h is the height of the prism. CCSS SENSE-MAKING Find the volume of 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is each cylinder. Round to the nearest tenth.

ANSWER: 58.14 ft3 16.

14. an oblique hexagonal prism with a height of 15 SOLUTION: centimeters and with a base area of 136 square r = 5 yd and h = 18 yd centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER:

area are same. 3 1413.7 yd

17. ANSWER: 3 SOLUTION: 2040 cm r = 6 cm and h = 3.6 cm.

15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 407.2 cm3

18. ANSWER: SOLUTION: 1534.25 in3 r = 5.5 in. CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. Use the Pythagorean Theorem to find the height of the cylinder.

16. SOLUTION: r = 5 yd and h = 18 yd

Now you can find the volume. ANSWER:

3 1413.7 yd

17. ANSWER: SOLUTION: 3 r = 6 cm and h = 3.6 cm. 823.0 in

19. SOLUTION: ANSWER: If two solids have the same height h and the same cross-sectional area B at every level, then they have 3 407.2 cm the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm.

18. SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder. ANSWER: 2686.1 mm3

SOLUTION: Now you can find the volume. The planter is to be filled inches below the top, so

ANSWER:

3 823.0 in

ANSWER: 3 2740.5 in

19. 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two SOLUTION: cylindrical candles. Each candle has a diameter of 9 If two solids have the same height h and the same centimeters and a height of 15 centimeters, as shown cross-sectional area B at every level, then they have at the right. What is the volume of the empty space the same volume. So, the volume of a right prism and in the box? an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm.

SOLUTION: The volume of the empty space is the difference of

volumes of the rectangular prism and the cylinders.

ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil ANSWER: in the planter if the planter is filled to inches 521.5 cm3 below the top? 22. SANDCASTLES In a sandcastle competition, SOLUTION: contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct The planter is to be filled inches below the top, so amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3 2740.5 in

21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 Therefore, the diameter of the cylinders should be centimeters and a height of 15 centimeters, as shown about 2.52 ft. at the right. What is the volume of the empty space in the box? ANSWER: 2.52 ft Find the volume of the solid formed by each net.

SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders. 23. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 ANSWER: cm. Use the Pythagorean Theorem to find the height 521.5 cm3 of the base.

22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct The height of the prism is 20 cm. amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. The volume V of a prism is V = Bh, where B is the What should the diameter of the cylinders be? area of the base, h is the height of the prism.

SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3934.9 cm3

Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER: 2.52 ft Find the volume of the solid formed by each 24. net. SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique.

23. The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m. SOLUTION:

The middle piece of the net is the front of the solid. If two solids have the same height h and the same The top and bottom pieces are the bases and the cross-sectional area B at every level, then they have pieces on the ends are the side faces. This is a the same volume. So, the volume of a right prism and

triangular prism. an oblique one of the same height and cross sectional area are same. One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm.

The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism. ANSWER: 3 48.9 m

25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the ANSWER: same radius, what is the height of the larger can? 3934.9 cm3

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius. 24. SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant

height is 6 m. The height of the new can will be 35.1 cm.

If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 35.1 cm the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 26. CHANGING DIMENSIONS A cylinder has a area are same. radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged.

SOLUTION: ANSWER: 3 48.9 m

25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% a. larger than the regular can. If both cans have the When the height is tripled, h = 3h. same radius, what is the height of the larger can?

SOLUTION: The volume of the smaller can is When the height is tripled, the volume is multiplied by 3. The volume of the new can is 130% of the smaller b. When the radius is tripled, r = 3r. can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

The height of the new can will be 35.1 cm. c. When the height and the radius are tripled, r = 3r and h = 3h. ANSWER: 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. When the height and the radius are tripled, the c. Both the radius and the height are tripled. volume is multiplied by 27. d. The dimensions are exchanged. SOLUTION: d. When the dimensions are exchanged, r = 8 and h = 5 cm.

a. When the height is tripled, h = 3h. Compare to the original volume.

The volume is multiplied by .

ANSWER: When the height is tripled, the volume is multiplied by a. The volume is multiplied by 3. 3. b. The volume is multiplied by 32 or 9. 3 b. When the radius is tripled, r = 3r. c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume. So, when the radius is tripled, the volume is multiplied by 9. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 c. When the height and the radius are tripled, r = 3r pounds, what is the soil’s bulk density? b. Assuming and h = 3h. that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION:

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h a. = 5 cm. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

Compare to the original volume. The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The volume is multiplied by . The soil density is thus:

ANSWER: a. The volume is multiplied by 3. 2 b. The volume is multiplied by 3 or 9. 3 c. The volume is multiplied by 3 or 27. b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. d. The volume is multiplied by c. 27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 ANSWER: pounds, what is the soil’s bulk density? b. Assuming 3 that all other factors are favorable, how well should a a. 0.0019 lb / in plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. b. The plant should grow well in this soil since the c. If a bag of this soil holds 2.5 cubic feet, what is its bulk density of 0.0019 lb / in3 is close to the desired 3 weight in pounds? bulk density of 0.0018 lb / in . SOLUTION: c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. 28. SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism

is 4 cm by 3 cm and the height is 5 cm. The weight of the soil is the weight of the pot with

soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus:

ANSWER: 225 cm3 b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

29.

SOLUTION: ANSWER: The solid is a combination of a rectangular prism and 3 a. 0.0019 lb / in a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular b. The plant should grow well in this soil since the prisms.

bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

ANSWER: 3 120 m

28. SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism 30. is 4 cm by 3 cm and the height is 5 cm. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

ANSWER: 225 cm3

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 29. centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of SOLUTION: the rim of the holder is 1 centimeter. What is the The solid is a combination of a rectangular prism and volume of the rubberized material that makes up the a right triangular prism. The total volume of the solid holder? is the sum of the volumes of the two rectangular prisms.

SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and ANSWER: the space used for the can. The container has a 3 120 m radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

30.

SOLUTION: The solid is a combination of a rectangular prism and two half cylinders. Therefore, the volume of the rubberized material is

about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? ANSWER: SOLUTION: 260.5 in3

holder?

ANSWER: 5.7 cm

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: SOLUTION: The volume of the rubberized material is the Use the surface area formula to solve for r. difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: The radius is 6. Find the volume. 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height?

SOLUTION:

ANSWER: 678.6 in3

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION:

Use the surface area formula to solve for r.

Find the volume.

ANSWER: 3 576 in

Find the difference between the volumes. ANSWER: 678.6 in3

ANSWER: 3,190,680.0 cm3

SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the ANSWER: rectangular prism is 6 ft by 10 ft and the height is 15 3 576 in ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height 35. ARCHITECTURE A cylindrical stainless steel of the trapezoidal prism is 15 ft. The total volume of column is used to hide a ventilation system in a new the solid is the sum of the volumes of the two building. According to the specifications, the diameter prisms. of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively. ANSWER:

3 Find the difference between the volumes. 1575 ft

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular SOLUTION: swimming pool is sloped so one end of the pool is 6 Find the volume of the original container. feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h. SOLUTION: The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms. ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3)

SOLUTION: ANSWER: Sample answers: 3 1575 ft For any cylindrical container, we have the following 37. CHANGING DIMENSIONS A soy milk company equation for volume: is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the same. What will be the height of the new containers?

The volume of the new container is 125% of the If we choose a height of say 4 in., then we can solve original container, with the same base dimensions. for the radius. Use 1.25V and B to find h.

ANSWER:

drawing. (1 cup ≈ 14.4375 in3) Choose numbers for any two of the dimensions and SOLUTION: we can solve for the third. Let l = 2.25 in. and w = Sample answers: 2.5 in.

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

ANSWER: Sample answers: Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

SOLUTION: The total cost for the second contractor is about . The base of the prism can be divided into 5 congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal Therefore, the second contractor is a less expensive prism is a combination of 5 triangular prisms of height option.

10 cm. Find the base area of each triangular prism. ANSWER:

Because 2.96 yd3 of concrete are needed, the

Therefore, the volume of the pentagonal prism is second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique ANSWER: cylinders 3 1100 cm ; Each triangular prism has a base area of . 2 a. GEOMETRIC Draw a right cylinder and an or 22 cm and a height of 10 cm. oblique cylinder with a height of 10 meters and a diameter of 6 meters. 40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 b. VERBAL A square prism has a height of 10 feet long, 12 feet wide, and 4 inches thick. One meters and a base edge of 6 meters. Is its volume contractor bid $2225 for the project. A second greater than, less than, or equal to the volume of contractor bid $500 per cubic yard for the new patio the cylinder? Explain. and $700 for removal of the old patio. Which is the less expensive option? Explain. c. ANALYTICAL Describe which change affects SOLUTION: the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. Convert all of the dimensions to yards. SOLUTION: 20 feet = yd a. The oblique cylinder should look like the right 12 feet = 4 yd cylinder (same height and size), except that it is pushed a little to the side, like a slinky. 4 in. = yd

Find the volume.

b. Find the volume of each.

The total cost for the second contractor is about

Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the The volume of the square prism is greater. second contractor is less expensive at $2181.50. c. Do each scenario.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain. Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater. c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the 2 height by x or multiplying the radius by x. Explain. For example, if x = 0.5, then x = 0.25, which is less than x. SOLUTION: a. The oblique cylinder should look like the right ANSWER: cylinder (same height and size), except that it is a. pushed a little to the side, like a slinky.

b. Find the volume of each. b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater.

42. CCSS CRITIQUE Franciso and Valerie each The volume of the square prism is greater. calculated the volume of an equilateral triangular c. Do each scenario. prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Assuming x > 1, multiplying the radius by x makes Francisco used the standard formula for the volume 2 the volume x times greater. of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the 2 For example, if x = 0.5, then x = 0.25, which is less base.

than x.

ANSWER: a. ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the b. Greater than; a square with a side length of 6 m container. Describe possible dimensions of the 2 container if it is each of the following shapes. has an area of 36 m . A circle with a diameter of 6 2 m has an area of 9π or 28.3 m . Since the heights a. rectangular prism are the same, the volume of the square prism is greater. b. square prism

c. Multiplying the radius by x ; since the volume is c. represented by π r 2 h, multiplying the height by x triangular prism with a right triangle as the base

makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular SOLUTION: prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your 3 reasoning. The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the container is 60π in 3.

a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. SOLUTION:

Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the 3 by 5 by 4π base.

b. Choose some basic values for 2 of the sides, and

then determine the third side. Base: 5 by 5.

ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the 5 by 5 by container if it is each of the following shapes.

c. Choose some basic values for 2 of the sides, and a. rectangular prism then determine the third side. Base: Legs: 3 by 4.

b. square prism

c. triangular prism with a right triangle as the base

3 by 4 by 10π

SOLUTION: ANSWER: Sample answers: 3 The volume of the can is 20π in . It takes three full a. 3 by 5 by 4π cans to fill the container, so the volume of the

container is 60π in 3. b. 5 by 5 by a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum.

I am new to gardening. The nursery will deliver a 3 by 5 by 4π truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of b. Choose some basic values for 2 of the sides, and soil? How do I know what to order? then determine the third side. Base: 5 by 5. SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine 5 by 5 by the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a c. Choose some basic values for 2 of the sides, and volume of 50 cubic centimeters. then determine the third side. Base: Legs: 3 by 4. SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest values to choose.

Sample answer:

3 by 4 by 10π

ANSWER: Sample answers: ANSWER: a. 3 by 5 by 4π Sample answer:

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 10π in. REASONING 44. WRITING IN MATH Write a helpful response to 46. Determine whether the following

the following question posted on an Internet statement is true or false. Explain. gardening forum. Two cylinders with the same height and the same

lateral area must have the same volume. I am new to gardening. The nursery will deliver a SOLUTION: truckload of soil, which they say is 4 yards. I True; if two cylinders have the same height (h = h ) know that a yard is 3 feet, but what is a yard of 1 2 soil? How do I know what to order? and the same lateral area (L1 = L2), the circular SOLUTION: bases must have the same area.

Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine The radii must also be equal. the number of cubic yards of soil needed. ANSWER: 45. OPEN ENDED Draw and label a prism that has a True; if two cylinders have the same height and the volume of 50 cubic centimeters. same lateral area, the circular bases must have the 2 SOLUTION: same area. Therefore, πr h is the same for each cylinder. Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest 47. WRITING IN MATH How are the formulas for values to choose. the volume of a prism and the volume of a cylinder similar? How are they different? Sample answer: SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a

cylinder is a circle, so its area is πr2. ANSWER: Sample answer: ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr . 46. REASONING Determine whether the following 48. The volume of a triangular prism is 1380 cubic statement is true or false. Explain. centimeters. Its base is a right triangle with legs Two cylinders with the same height and the same measuring 8 centimeters and 15 centimeters. What is

lateral area must have the same volume. the height of the prism? A 34.5 cm SOLUTION: B 23 cm True; if two cylinders have the same height (h1 = h2) C 17 cm D 11.5 cm and the same lateral area (L1 = L2), the circular bases must have the same area. SOLUTION:

The radii must also be equal.

ANSWER: ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the B 2 same area. Therefore, πr h is the same for each 49. A cylindrical tank used for oil storage has a height cylinder. that is half the length of its radius. If the volume of the tank is 1,122,360 ft3, what is the tank’s radius? 47. WRITING IN MATH How are the formulas for F 89.4 ft the volume of a prism and the volume of a cylinder similar? How are they different? G 178.8 ft H 280.9 ft SOLUTION: J 561.8 ft Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, SOLUTION: so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? ANSWER: A 34.5 cm F B 23 cm C 17 cm 50. SHORT RESPONSE What is the ratio of the area D 11.5 cm of the circle to the area of the square? SOLUTION:

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER: 12-4ANSWER: Volumes of Prisms and Cylinders B

49. A cylindrical tank used for oil storage has a height 51. SAT/ACT A county proposes to enact a new 0.5% that is half the length of its radius. If the volume of property tax. What would be the additional tax the tank is 1,122,360 ft3, what is the tank’s radius? amount for a landowner whose property has a F 89.4 ft taxable value of $85,000? G 178.8 ft A $4.25 H 280.9 ft B $170 J 561.8 ft C $425 D $4250 SOLUTION: E $42,500 SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

ANSWER: F 52. 50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square? SOLUTION: The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. SOLUTION: The radius of the circle is 2x and the length of each The slant height is the height of each of the side of the square is 4x. So, the ratio of the areas can congruent lateral triangular faces. Use the

be written as shown. Pythagorean Theorem to find the slant height.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 eSolutionsD $4250Manual - Powered by Cognero Page 19 E $42,500 SOLUTION: Find the 0.5% of $85,000.

Find the perimeter and area of the equilateral Therefore, the correct choice is C. triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. ANSWER:

C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52.

SOLUTION: The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the The perimeter is P = 3 × 10 or 30 feet. Pythagorean Theorem to find the slant height.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 212.1 ft2; 255.4 ft2

53. SOLUTION: The perimeter is P = 3 × 10 or 30 feet. The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the

regular pyramid. 2 So, the lateral area of the pyramid is 126 cm .

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is about 212.1 ft2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: Therefore, the surface area of the pyramid is about 2 2 2 126 cm ; 175 cm 255.4 ft .

ANSWER: 212.1 ft2; 255.4 ft2

54. SOLUTION: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. 53. A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

Use a trigonometric ratio to find the measure of the apothem a.

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base. Find the lateral area and surface area of the pyramid.

Therefore, the surface area of the pyramid is 175 2 cm2. So, the lateral area of the pyramid is 472.5 in .

ANSWER: 2 2 126 cm ; 175 cm

54. Therefore, the surface area of the pyramid is about SOLUTION: 758.9 in2. The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the ANSWER: 2 2 angle formed in the triangle below is 30°. 472.5 in ; 758.9 in

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

SOLUTION: Use a trigonometric ratio to find the measure of the The area that needs to be coated is the sum of the apothem a. lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 2 205 in

Find the lateral area and surface area of the Find the indicated measure. Round to the nearest tenth. pyramid. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

So, the lateral area of the pyramid is 472.5 in2.

The diameter of the circle is about 8.3 m.

ANSWER: Therefore, the surface area of the pyramid is about 8.3 m 758.9 in2. 57. Find the diameter of a circle with an area of 102 square centimeters. ANSWER: SOLUTION: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

The diameter of the circle is about 11.4 m. SOLUTION: The area that needs to be coated is the sum of the ANSWER: lateral area and one base area. 11.4 cm Therefore, the area that needs to be coated is 2(13 + 2 58. The area of a circle is 191 square feet. Find the 9)(2) + 13(9) = 205 in . radius. ANSWER: SOLUTION: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches.

SOLUTION: The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

The diameter of the circle is about 11.4 m.

ANSWER:

11.4 cm 60. SOLUTION: 58. The area of a circle is 191 square feet. Find the The area A of a kite is one half the product of the radius. lengths of its diagonals, d1 and d2. SOLUTION: d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 ANSWER: 120 in 7.8 ft 59. Find the radius of a circle with an area of 271 square inches. SOLUTION: 61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or ANSWER: 2 kite. 378 m

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

Find the volume of each prism.

3 The volume is 108 cm . ANSWER:

26.95 m3 ANSWER: 3 4. an oblique pentagonal prism with a base area of 42 108 cm square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 5. 3 396 in SOLUTION: 3. the oblique rectangular prism shown at the right

SOLUTION: If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have 206.4 ft3 the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: 3 area are same. 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION:

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth. ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches 5. SOLUTION: SOLUTION:

ANSWER: 3 ANSWER: 410.1 in 3 206.4 ft 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 6. B 6400 SOLUTION: C 30,000 D If two solids have the same height h and the same 48,000 cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D. ANSWER: ANSWER: 3 D 1357.2 m CCSS SENSE-MAKING Find the volume of 7. a cylinder with a diameter of 16 centimeters and a each prism. height of 5.1 centimeters SOLUTION:

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches ANSWER: SOLUTION: 30 in3

7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000 D 48,000 ANSWER: 3 SOLUTION: 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) 12. = 48,000. SOLUTION: Therefore, the correct choice is D. The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. ANSWER: D Use the Pythagorean Theorem to find the height of the base. CCSS SENSE-MAKING Find the volume of each prism.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

The height of the prism is 6 cm.

ANSWER: ANSWER: 30 in3 324 cm3

11. 13. SOLUTION: SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the If two solids have the same height h and the same prism is 14 m. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 2 3 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is 539 m

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15

12. centimeters and with a base area of 136 square SOLUTION: centimeters The base is a right triangle with a leg length of 9 cm SOLUTION:

and the hypotenuse of length 15 cm. If two solids have the same height h and the same cross-sectional area B at every level, then they have Use the Pythagorean Theorem to find the height of the same volume. So, the volume of a right prism and the base. an oblique one of the same height and cross sectional area are same.

ANSWER: 3 2040 cm The height of the prism is 6 cm. 15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and 324 cm3 an oblique one of the same height and cross sectional area are same.

13. SOLUTION: ANSWER: If two solids have the same height h and the same 1534.25 in3 cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and CCSS SENSE-MAKING Find the volume of an oblique one of the same height and cross sectional each cylinder. Round to the nearest tenth. area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 16.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is SOLUTION: r = 5 yd and h = 18 yd

ANSWER: 58.14 ft3

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square ANSWER: centimeters 3 1413.7 yd SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a ANSWER: height of 17 inches 407.2 cm3 SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 18.

SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder.

ANSWER: 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16. SOLUTION:

r = 5 yd and h = 18 yd Now you can find the volume.

ANSWER: 3 1413.7 yd ANSWER: 3 823.0 in

17. SOLUTION: 19. r = 6 cm and h = 3.6 cm. SOLUTION:

r = 7.5 mm and h = 15.2 mm. ANSWER: 407.2 cm3

18. ANSWER: 3 SOLUTION: 2686.1 mm r = 5.5 in. 20. PLANTER A planter is in the shape of a Use the Pythagorean Theorem to find the height of rectangular prism 18 inches long, inches deep, the cylinder. and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER: 3 2740.5 in

ANSWER: 21. SHIPPING A box 18 centimeters by 9 centimeters 3 823.0 in by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box?

ANSWER: 2686.1 mm3 ANSWER: 521.5 cm3 20. PLANTER A planter is in the shape of a 22. SANDCASTLES In a sandcastle competition, rectangular prism 18 inches long, inches deep, contestants are allowed to use only water, shovels, and 12 inches high. What is the volume of potting soil and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that in the planter if the planter is filled to inches are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? below the top? SOLUTION: SOLUTION:

3 The planter is to be filled inches below the top, so V = 10 ft and h = 2 ft Use the formula to find r.

23. SOLUTION: The middle piece of the net is the front of the solid. SOLUTION: The top and bottom pieces are the bases and the The volume of the empty space is the difference of pieces on the ends are the side faces. This is a volumes of the rectangular prism and the cylinders. triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm. ANSWER: The volume V of a prism is V = Bh, where B is the 3 521.5 cm area of the base, h is the height of the prism.

3 V = 10 ft and h = 2 ft Use the formula to find r.

24. Therefore, the diameter of the cylinders should be SOLUTION: about 2.52 ft. The circular bases at the top and bottom of the net ANSWER: indicate that this is a cylinder. If the middle piece 2.52 ft were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it Find the volume of the solid formed by each is oblique. net. The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 23. an oblique one of the same height and cross sectional area are same. SOLUTION: The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base. ANSWER:

3 48.9 m

The height of the prism is 20 cm. 25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 The volume V of a prism is V = Bh, where B is the centimeters. A new can is advertised as being 30% area of the base, h is the height of the prism. larger than the regular can. If both cans have the same radius, what is the height of the larger can?

ANSWER: SOLUTION: 3 3934.9 cm The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

24.

SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece The height of the new can will be 35.1 cm. were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it ANSWER: is oblique. 35.1 cm

The radius is 1.8 m, the height is 4.8 m, and the slant 26. CHANGING DIMENSIONS A cylinder has a height is 6 m. radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the If two solids have the same height h and the same cylinder. cross-sectional area B at every level, then they have a. The height is tripled. the same volume. So, the volume of a right prism and b. The radius is tripled. an oblique one of the same height and cross sectional c. Both the radius and the height are tripled. area are same. d. The dimensions are exchanged.

SOLUTION:

a. ANSWER: When the height is tripled, h = 3h. 3 48.9 m

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm.

ANSWER:

35.1 cm When the height and the radius are tripled, the CHANGING DIMENSIONS 26. A cylinder has a volume is multiplied by 27. radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. d. When the dimensions are exchanged, r = 8 and h = 5 cm. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged.

SOLUTION:

Compare to the original volume.

a. The volume is multiplied by . When the height is tripled, h = 3h. ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

So, when the radius is tripled, the volume is multiplied per square inch is desirable for root growth? Explain. by 9. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? c. When the height and the radius are tripled, r = 3r SOLUTION: and h = 3h.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus:

Compare to the original volume.

3 3 The volume is multiplied by . b. 0.0018 lb/in is close to 0.0019 lb/in so the plant should grow fairly well.

ANSWER: c. a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific ANSWER: plant will grow in it. The density 3 a. 0.0019 lb / in of the soil sample is the ratio of its weight to its

volume. b. The plant should grow well in this soil since the 3 a. If the weight of the container with the soil is 20 bulk density of 0.0019 lb / in is close to the desired 3 pounds and the weight of the container alone is 5 bulk density of 0.0018 lb / in . pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a c. 8.3 lb plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. Find the volume of each composite solid. Round c. If a bag of this soil holds 2.5 cubic feet, what is its to the nearest tenth if necessary. weight in pounds? SOLUTION:

28. SOLUTION:

The solid is a combination of two rectangular prisms. a. First calculate the volume of soil in the pot. Then The base of one rectangular prism is 5 cm by 3 cm divide the weight of the soil by the volume. and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

The weight of the soil is the weight of the pot with soil minus the weight of the pot. ANSWER: W = 20 – 5 = 15 lbs. 3 225 cm

The soil density is thus:

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

29. c. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 a. 0.0019 lb / in

b. The plant should grow well in this soil since the 3 bulk density of 0.0019 lb / in is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb ANSWER: 3 120 m Find the volume of each composite solid. Round to the nearest tenth if necessary.

30. 28. SOLUTION: The solid is a combination of a rectangular prism and SOLUTION: two half cylinders. The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

ANSWER: ANSWER: 3 225 cm3 260.5 in 31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular

prisms. SOLUTION: The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the ANSWER: radius of the base and h is the height of the cylinder. 3 120 m

Therefore, the volume of the rubberized material is about 304.1 cm3. 30. ANSWER: SOLUTION: 3 304.1 cm The solid is a combination of a rectangular prism and two half cylinders. Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? SOLUTION:

ANSWER: 260.5 in3

holder? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER:

3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic ANSWER: centimeters. The diameter of the can is 9 3 678.6 in centimeters. What is the height? SOLUTION: 34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER: 5.7 cm Find the volume. 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

The volume will be the highest when the diameter is The radius is 6. Find the volume. 95 cm and will be the lowest when it is 30 cm.That is

when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 678.6 in3

34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 ANSWER: inches. What is the volume? 3,190,680.0 cm3 SOLUTION: 36. CCSS MODELING The base of a rectangular Use the surface area formula to find the length of the swimming pool is sloped so one end of the pool is 6

base of the prism. feet deep and the other end is 3 feet deep, as shown

in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest ANSWER: 3 tenth cubic centimeter. 1575 ft SOLUTION: 37. CHANGING DIMENSIONS A soy milk company The volume will be the highest when the diameter is is planning a promotion in which the volume of soy 95 cm and will be the lowest when it is 30 cm.That is milk in each container will be increased by 25%. The when the radii are 47.5 cm and 15 cm respectively. company wants the base of the container to stay the

same. What will be the height of the new containers? Find the difference between the volumes.

SOLUTION: Find the volume of the original container.

ANSWER: 3,190,680.0 cm3

SOLUTION: The swimming pool is a combination of a rectangular ANSWER: prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of 38. DESIGN Sketch and label (in inches) three different the solid is the sum of the volumes of the two designs for a dry ingredient measuring cup that holds prisms. 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

ANSWER: 3 1575 ft

SOLUTION: Find the volume of the original container. If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

For any rectangular container, the volume equation is:

ANSWER:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in. 38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius. ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of ANSWER: 2 Sample answers: or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

Therefore, the second contractor is a less expensive option.

ANSWER: b. VERBAL A square prism has a height of 10 3 1100 cm ; Each triangular prism has a base area of meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of 2 or 22 cm and a height of 10 cm. the cylinder? Explain.

40. PATIOS Mr. Thomas is planning to remove an old c. ANALYTICAL Describe which change affects patio and install a new rectangular concrete patio 20 the volume of the cylinder more: multiplying the feet long, 12 feet wide, and 4 inches thick. One height by x or multiplying the radius by x. Explain. contractor bid $2225 for the project. A second SOLUTION: contractor bid $500 per cubic yard for the new patio a. The oblique cylinder should look like the right and $700 for removal of the old patio. Which is the cylinder (same height and size), except that it is less expensive option? Explain. pushed a little to the side, like a slinky. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd b. Find the volume of each. Find the volume.

The total cost for the second contractor is about

. The volume of the square prism is greater. c. Do each scenario. Therefore, the second contractor is a less expensive option.

ANSWER: 3 Because 2.96 yd of concrete are needed, the second contractor is less expensive at $2181.50.

b. VERBAL A square prism has a height of 10 2 For example, if x = 0.5, then x = 0.25, which is less meters and a base edge of 6 meters. Is its volume than x. greater than, less than, or equal to the volume of the cylinder? Explain. ANSWER: a. c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky. b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater.

The volume of the square prism is greater. c. Do each scenario.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater. ANSWER: For example, if x = 0.5, then x2 = 0.25, which is less Francisco; Valerie incorrectly used as the than x. length of one side of the triangular base. Francisco ANSWER: used a different approach, but his solution is correct. a. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 c. triangular prism with a right triangle as the base

m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater.

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x SOLUTION: makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, 3 assuming x > 1. The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the 3 42. CCSS CRITIQUE Franciso and Valerie each container is 60π in . calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 a. Choose some basic values for 2 of the sides, and units. Is either of them correct? Explain your then determine the third side. Base: 3 by 5. reasoning.

SOLUTION: Francisco; Valerie incorrectly used as the 3 by 5 by 4π length of one side of the triangular base. Francisco used a different approach, but his solution is correct. b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

ANSWER: Francisco; Valerie incorrectly used as the 5 by 5 by length of one side of the triangular base. Francisco used a different approach, but his solution is correct. c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism 3 by 4 by 10π c. triangular prism with a right triangle as the base ANSWER:

Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by SOLUTION:

c. base with legs measuring 3 in. and 4 in., height 3 The volume of the can is 20π in . It takes three full 10π in. cans to fill the container, so the volume of the container is 60π in 3. 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum. a. Choose some basic values for 2 of the sides, and

then determine the third side. Base: 3 by 5. I am new to gardening. The nursery will deliver a

truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your 3 by 5 by 4π garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. SOLUTION:

Choose 3 values that multiply to make 50. The 5 by 5 by factors of 50 are 2, 5, 5, so these are the simplest values to choose. c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. Sample answer:

ANSWER: Sample answer: 3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π 46. REASONING Determine whether the following statement is true or false. Explain. b. 5 by 5 by Two cylinders with the same height and the same lateral area must have the same volume. c. base with legs measuring 3 in. and 4 in., height 10π in. SOLUTION: True; if two cylinders have the same height (h1 = h2) 44. WRITING IN MATH Write a helpful response to and the same lateral area (L = L ), the circular the following question posted on an Internet 1 2 gardening forum. bases must have the same area.

garden in cubic feet and divide by 27 to determine The radii must also be equal. the number of cubic yards of soil needed. ANSWER: ANSWER: True; if two cylinders have the same height and the Sample answer: The nursery means a cubic yard, same lateral area, the circular bases must have the 2 which is 33 or 27 cubic feet. Find the volume of your same area. Therefore, πr h is the same for each garden in cubic feet and divide by 27 to determine cylinder. the number of cubic yards of soil needed. 47. WRITING IN MATH How are the formulas for 45. OPEN ENDED Draw and label a prism that has a the volume of a prism and the volume of a cylinder volume of 50 cubic centimeters. similar? How are they different? SOLUTION: SOLUTION: Choose 3 values that multiply to make 50. The Both formulas involve multiplying the area of the factors of 50 are 2, 5, 5, so these are the simplest base by the height. The base of a prism is a polygon, values to choose. so the expression representing the area varies, depending on the type of polygon it is. The base of a 2 Sample answer: cylinder is a circle, so its area is πr . ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 ANSWER: base of a cylinder is a circle, so its area is πr . Sample answer: 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm C 17 cm 46. REASONING Determine whether the following D statement is true or false. Explain. 11.5 cm Two cylinders with the same height and the same SOLUTION: lateral area must have the same volume. SOLUTION:

True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

ANSWER: B

49. A cylindrical tank used for oil storage has a height The radii must also be equal. that is half the length of its radius. If the volume of the tank is 1,122,360 ft3, what is the tank’s radius? ANSWER: F 89.4 ft True; if two cylinders have the same height and the G 178.8 ft same lateral area, the circular bases must have the 2 H 280.9 ft same area. Therefore, πr h is the same for each J 561.8 ft cylinder. SOLUTION: 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

48. The volume of a triangular prism is 1380 cubic 50. SHORT RESPONSE What is the ratio of the area centimeters. Its base is a right triangle with legs of the circle to the area of the square? measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm C 17 cm SOLUTION: D 11.5 cm The radius of the circle is 2x and the length of each SOLUTION: side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax

amount for a landowner whose property has a ANSWER: taxable value of $85,000? A B $4.25 B $170 49. A cylindrical tank used for oil storage has a height C $425 that is half the length of its radius. If the volume of D $4250 the tank is 1,122,360 ft3, what is the tank’s radius? E $42,500 F 89.4 ft SOLUTION: G 178.8 ft Find the 0.5% of $85,000. H 280.9 ft J 561.8 ft SOLUTION: Therefore, the correct choice is C. ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , ANSWER: F where is the slant height and P is the perimeter of the base. 50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square? The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25 B $170 C $425 D $4250 Find the perimeter and area of the equilateral E $42,500 triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION: The perimeter is P = 3 × 10 or 30 feet. The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

12-4 Volumes of Prisms and Cylinders 2 So, the lateral area of the pyramid is about 212.1 ft .

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem Therefore, the surface area of the pyramid is about 2 to find the height h of the triangle. 255.4 ft .

ANSWER: 212.1 ft2; 255.4 ft2

53.

2 So, the lateral area of the pyramid is 126 cm .

So, the area of the base B is ft2.

The surface area S of a regular

Find the lateral area L and surface area S of the pyramid is , where regular pyramid. L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is about 212.1 ft2. Therefore, the surface area of the pyramid is 175 cm2. eSolutions Manual - Powered by Cognero ANSWER: Page 20 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 54. 2 255.4 ft . SOLUTION: ANSWER: The pyramid has a slant height of 15 inches and the 2 2 base is a hexagon with sides of 10.5 inches. 212.1 ft ; 255.4 ft A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

53. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Use a trigonometric ratio to find the measure of the Here, the base is a square of side 7 cm and the slant apothem a. height is 9 cm.

So, the lateral area of the pyramid is 126 cm2. Find the lateral area and surface area of the pyramid.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 2 2 54. 472.5 in ; 758.9 in SOLUTION: 55. BAKING Many baking pans are given a special The pyramid has a slant height of 15 inches and the nonstick coating. A rectangular cake pan is 9 inches base is a hexagon with sides of 10.5 inches. by 13 inches by 2 inches deep. What is the area of A central angle of the hexagon is or 60°, so the the inside of the pan that needs to be coated? angle formed in the triangle below is 30°.

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: Use a trigonometric ratio to find the measure of the 2 apothem a. 205 in Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter.

SOLUTION:

Find the lateral area and surface area of the pyramid.

So, the lateral area of the pyramid is 472.5 in2. The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters. SOLUTION:

Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of The diameter of the circle is about 11.4 m. the inside of the pan that needs to be coated? ANSWER: 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION: SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. ANSWER: 7.8 ft SOLUTION: 59. Find the radius of a circle with an area of 271 square inches. SOLUTION:

The diameter of the circle is about 8.3 m.

ANSWER:

8.3 m

57. Find the diameter of a circle with an area of 102 ANSWER: square centimeters. 9.3 in. SOLUTION: Find the area of each trapezoid, rhombus, or kite.

60.

SOLUTION: The area A of a kite is one half the product of the The diameter of the circle is about 11.4 m. lengths of its diagonals, d1 and d2.

ANSWER: d = 12 in. and d = 7 + 13 = 20 in. 11.4 cm 1 2

58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 2 120 in

ANSWER: 2 378 m

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right

Find the volume of each prism.

3 The volume is 108 cm . ANSWER: 26.95 m3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: 5. 3 396 in SOLUTION:

3. the oblique rectangular prism shown at the right

6. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder ANSWER: and an oblique one of the same height and cross sectional area are same. 3 26.95 m 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and ANSWER: an oblique one of the same height and cross sectional 3 area are same. 1357.2 m

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters SOLUTION:

ANSWER: 3 218.4 cm

Find the volume of each cylinder. Round to the ANSWER: nearest tenth. 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches 5. SOLUTION: SOLUTION:

ANSWER: 3 410.1 in ANSWER: 3 MULTIPLE CHOICE 206.4 ft 9. A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6. 6400 C 30,000 SOLUTION: D 48,000 If two solids have the same height h and the same cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches ANSWER: SOLUTION: 30 in3

B 6400 C 30,000 D 48,000 ANSWER: 3 SOLUTION: 539 m

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) 12. = 48,000. SOLUTION:

The base is a right triangle with a leg length of 9 cm Therefore, the correct choice is D. and the hypotenuse of length 15 cm. ANSWER: D Use the Pythagorean Theorem to find the height of the base. CCSS SENSE-MAKING Find the volume of each prism.

10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. The height of the prism is 6 cm.

ANSWER: ANSWER: 3 30 in 324 cm3

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 ANSWER: B = 11.4 ft and h = 5.1 ft. Therefore, the volume is 3 539 m

ANSWER: 58.14 ft3

ANSWER: 3 2040 cm

The height of the prism is 6 cm. 15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and 3 an oblique one of the same height and cross sectional 324 cm area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 16.

SOLUTION: 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is r = 5 yd and h = 18 yd

ANSWER: 58.14 ft3

ANSWER: 3 2040 cm

Use the Pythagorean Theorem to find the height of

the cylinder.

ANSWER: 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth.

16.

SOLUTION: r = 5 yd and h = 18 yd Now you can find the volume.

ANSWER: 3 ANSWER: 1413.7 yd 3 823.0 in

r = 7.5 mm and h = 15.2 mm. ANSWER: 407.2 cm3

ANSWER: 18. 2686.1 mm3 SOLUTION: r = 5.5 in. 20. PLANTER A planter is in the shape of a

Use the Pythagorean Theorem to find the height of rectangular prism 18 inches long, inches deep,

the cylinder. and 12 inches high. What is the volume of potting soil

in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

Now you can find the volume.

ANSWER: 3 2740.5 in

ANSWER: ANSWER: 2686.1 mm3 521.5 cm3 20. PLANTER A planter is in the shape of a 22. SANDCASTLES In a sandcastle competition, rectangular prism 18 inches long, inches deep, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct and 12 inches high. What is the volume of potting soil amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. in the planter if the planter is filled to inches What should the diameter of the cylinders be? below the top? SOLUTION: SOLUTION: 3 The planter is to be filled inches below the top, so V = 10 ft and h = 2 ft Use the formula to find r.

Therefore, the diameter of the cylinders should be about 2.52 ft. ANSWER: 3 2740.5 in ANSWER: 2.52 ft 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two Find the volume of the solid formed by each cylindrical candles. Each candle has a diameter of 9 net. centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box?

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm.

ANSWER: The volume V of a prism is V = Bh, where B is the 521.5 cm3 area of the base, h is the height of the prism.

3 V = 10 ft and h = 2 ft Use the formula to find r.

24.

One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base. ANSWER: 3 48.9 m

ANSWER: SOLUTION: 3934.9 cm3 The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

24. SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece The height of the new can will be 35.1 cm. were a rectangle, then the prism would be right. ANSWER: However, since the middle piece is a parallelogram, it is oblique. 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a The radius is 1.8 m, the height is 4.8 m, and the slant radius of 5 centimeters and a height of 8 centimeters.

height is 6 m. Describe how each change affects the volume of the cylinder. If two solids have the same height h and the same a. The height is tripled. cross-sectional area B at every level, then they have b. The radius is tripled. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional c. Both the radius and the height are tripled. area are same. d. The dimensions are exchanged. SOLUTION:

a. ANSWER: When the height is tripled, h = 3h. 3 48.9 m

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius. So, when the radius is tripled, the volume is multiplied by 9.

c. When the height and the radius are tripled, r = 3r and h = 3h.

The height of the new can will be 35.1 cm. ANSWER: 35.1 cm When the height and the radius are tripled, the 26. CHANGING DIMENSIONS A cylinder has a volume is multiplied by 27. radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the d. When the dimensions are exchanged, r = 8 and h cylinder. = 5 cm. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged. SOLUTION:

Compare to the original volume.

The volume is multiplied by . a.

When the height is tripled, h = 3h. ANSWER:

a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

When the height is tripled, the volume is multiplied by 27. SOIL A soil scientist wants to determine the bulk 3. density of a potting soil to assess how well a specific plant will grow in it. The density b. When the radius is tripled, r = 3r. of the soil sample is the ratio of its weight to its

volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. So, when the radius is tripled, the volume is multiplied c. If a bag of this soil holds 2.5 cubic feet, what is its by 9. weight in pounds?

SOLUTION: c. When the height and the radius are tripled, r = 3r and h = 3h.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h

= 5 cm. The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus:

Compare to the original volume.

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant The volume is multiplied by . should grow fairly well.

ANSWER: c. a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

SOIL 27. A soil scientist wants to determine the bulk ANSWER: density of a potting soil to assess how well a specific 3 plant will grow in it. The density a. 0.0019 lb / in of the soil sample is the ratio of its weight to its volume. b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired a. If the weight of the container with the soil is 20 3 bulk density of 0.0018 lb / in . pounds and the weight of the container alone is 5

pounds, what is the soil’s bulk density? b. Assuming c. 8.3 lb that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound Find the volume of each composite solid. Round per square inch is desirable for root growth? Explain. to the nearest tenth if necessary. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION:

The weight of the soil is the weight of the pot with ANSWER: soil minus the weight of the pot. 3 W = 20 – 5 = 15 lbs. 225 cm

The soil density is thus:

is the sum of the volumes of the two rectangular prisms.

ANSWER: 3 a. 0.0019 lb / in

30. SOLUTION: 28. The solid is a combination of a rectangular prism and SOLUTION: two half cylinders. The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

29. SOLUTION: The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular

prisms. SOLUTION:

The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius

of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder. ANSWER: 3 120 m

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 260.5 in3

MANUFACTURING 31. A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the ANSWER: thickness of the base of the holder. The thickness of 5.7 cm the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

The radius is 6. Find the volume.

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm Find each measure to the nearest tenth. ANSWER: 32. A cylindrical can has a volume of 363 cubic 3 centimeters. The diameter of the can is 9 678.6 in centimeters. What is the height? 34. A rectangular prism has a surface area of 432 SOLUTION: square inches, a height of 6 inches, and a width of 12 inches. What is the volume? SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER:

5.7 cm Find the volume.

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

ANSWER: 3 576 in

SOLUTION: The volume will be the highest when the diameter is The radius is 6. Find the volume. 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 3 678.6 in

34. A rectangular prism has a surface area of 432 ANSWER: square inches, a height of 6 inches, and a width of 12 3 inches. What is the volume? 3,190,680.0 cm SOLUTION: 36. CCSS MODELING The base of a rectangular Use the surface area formula to find the length of the swimming pool is sloped so one end of the pool is 6 base of the prism. feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter

of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest ANSWER: and smallest possible column? Round to the nearest 3 tenth cubic centimeter. 1575 ft SOLUTION: 37. CHANGING DIMENSIONS A soy milk company The volume will be the highest when the diameter is is planning a promotion in which the volume of soy 95 cm and will be the lowest when it is 30 cm.That is milk in each container will be increased by 25%. The when the radii are 47.5 cm and 15 cm respectively. company wants the base of the container to stay the same. What will be the height of the new containers? Find the difference between the volumes.

SOLUTION:

Find the volume of the original container.

ANSWER: 3,190,680.0 cm3

in the figure. If the width is 15 feet, find the volume Use 1.25V and B to find h.

of water it takes to fill the pool.

SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

ANSWER: 3 1575 ft

37. CHANGING DIMENSIONS A soy milk company

is planning a promotion in which the volume of soy The last equation gives us a relation between the milk in each container will be increased by 25%. The radius and height of the cylinder that must be fulfilled company wants the base of the container to stay the to get the desired volume. First, choose a suitable same. What will be the height of the new containers? radius, say 1.85 in, and solve for the height.

SOLUTION: Find the volume of the original container. If we choose a height of say 4 in., then we can solve for the radius.

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

For any rectangular container, the volume equation is:

SOLUTION: Sample answers:

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius. ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 ANSWER: or 22 cm and a height of 10 cm. Sample answers:

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

Therefore, the second contractor is a less expensive option.

ANSWER: b. VERBAL A square prism has a height of 10 3 meters and a base edge of 6 meters. Is its volume 1100 cm ; Each triangular prism has a base area of greater than, less than, or equal to the volume of 2 the cylinder? Explain. or 22 cm and a height of 10 cm.

20 feet = yd 12 feet = 4 yd

4 in. = yd b. Find the volume of each.

Find the volume.

The total cost for the second contractor is about The volume of the square prism is greater. . c. Do each scenario.

Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the

second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders . a. GEOMETRIC Draw a right cylinder and an Assuming x > 1, multiplying the radius by x makes oblique cylinder with a height of 10 meters and a 2 diameter of 6 meters. the volume x times greater.

2 b. VERBAL A square prism has a height of 10 For example, if x = 0.5, then x = 0.25, which is less meters and a base edge of 6 meters. Is its volume than x. greater than, less than, or equal to the volume of the cylinder? Explain. ANSWER: a. c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. SOLUTION: a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky. b. Greater than; a square with a side length of 6 m 2 has an area of 36 m . A circle with a diameter of 6 m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is greater.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

Assuming x > 1, multiplying the radius by x makes 2 the volume x times greater. ANSWER: For example, if x = 0.5, then x2 = 0.25, which is less Francisco; Valerie incorrectly used as the than x. length of one side of the triangular base. Francisco used a different approach, but his solution is correct. ANSWER: a. 43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes.

a. rectangular prism

b. square prism

c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x SOLUTION: makes the volume x times greater. Multiplying the 3 radius by x makes the volume x 2 times greater, The volume of the can is 20π in . It takes three full assuming x > 1. cans to fill the container, so the volume of the container is 60π in 3. 42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular a. prism with an apothem of 4 units and height of 5 Choose some basic values for 2 of the sides, and

units. Is either of them correct? Explain your then determine the third side. Base: 3 by 5. reasoning.

SOLUTION: 3 by 5 by 4π Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco b. Choose some basic values for 2 of the sides, and

used a different approach, but his solution is correct. then determine the third side. Base: 5 by 5.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

a. rectangular prism

b. square prism 3 by 4 by 10π

c. triangular prism with a right triangle as the base ANSWER: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by SOLUTION: c. base with legs measuring 3 in. and 4 in., height 3 The volume of the can is 20π in . It takes three full 10π in. cans to fill the container, so the volume of the 44. WRITING IN MATH Write a helpful response to 3 container is 60π in . the following question posted on an Internet gardening forum. a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your 3 by 5 by 4π garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. b. Choose some basic values for 2 of the sides, and ANSWER: then determine the third side. Base: 5 by 5. Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

c. Choose some basic values for 2 of the sides, and Sample answer: then determine the third side. Base: Legs: 3 by 4.

ANSWER: Sample answer:

3 by 4 by 10π

ANSWER: Sample answers:

a. 3 by 5 by 4π 46. REASONING Determine whether the following statement is true or false. Explain. b. 5 by 5 by Two cylinders with the same height and the same lateral area must have the same volume. c. base with legs measuring 3 in. and 4 in., height SOLUTION: 10π in. True; if two cylinders have the same height (h1 = h2) 44. WRITING IN MATH Write a helpful response to and the same lateral area (L1 = L2), the circular the following question posted on an Internet bases must have the same area. gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your The radii must also be equal. garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. ANSWER: True; if two cylinders have the same height and the ANSWER: same lateral area, the circular bases must have the 2 Sample answer: The nursery means a cubic yard, same area. Therefore, πr h is the same for each 3 which is 3 or 27 cubic feet. Find the volume of your cylinder. garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder 45. OPEN ENDED Draw and label a prism that has a similar? How are they different? volume of 50 cubic centimeters. SOLUTION: SOLUTION: Both formulas involve multiplying the area of the Choose 3 values that multiply to make 50. The base by the height. The base of a prism is a polygon, factors of 50 are 2, 5, 5, so these are the simplest so the expression representing the area varies, values to choose. depending on the type of polygon it is. The base of a 2 cylinder is a circle, so its area is πr . Sample answer: ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr . ANSWER: Sample answer: 48. The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm C 17 cm 46. REASONING Determine whether the following D 11.5 cm statement is true or false. Explain. Two cylinders with the same height and the same SOLUTION: lateral area must have the same volume. SOLUTION:

True; if two cylinders have the same height (h1 = h2) and the same lateral area (L1 = L2), the circular bases must have the same area.

cylinder. SOLUTION: 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area ANSWER: varies, depending on the type of polygon it is. The 2 F base of a cylinder is a circle, so its area is πr . 50. SHORT RESPONSE What is the ratio of the area 48. The volume of a triangular prism is 1380 cubic of the circle to the area of the square? centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm C 17 cm SOLUTION: D 11.5 cm The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can SOLUTION: be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? ANSWER: A $4.25 B B $170 C $425 49. A cylindrical tank used for oil storage has a height D that is half the length of its radius. If the volume of $4250 E $42,500 the tank is 1,122,360 ft3, what is the tank’s radius? F 89.4 ft SOLUTION: G 178.8 ft Find the 0.5% of $85,000. H 280.9 ft J 561.8 ft Therefore, the correct choice is C. SOLUTION: ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52. SOLUTION:

The lateral area L of a regular pyramid is , ANSWER: where is the slant height and P is the perimeter of F the base.

SHORT RESPONSE 50. What is the ratio of the area The slant height is the height of each of the

of the circle to the area of the square? congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25

B $170 C $425 Find the perimeter and area of the equilateral D $4250 triangle for the base. Use the Pythagorean Theorem E $42,500 to find the height h of the triangle. SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52.

SOLUTION: The perimeter is P = 3 × 10 or 30 feet. The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the

congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

53. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm. The perimeter is P = 3 × 10 or 30 feet.

So, the lateral area of the pyramid is 126 cm2.

2 Therefore, the surface area of the pyramid is 175 So, the lateral area of the pyramid is about 212.1 ft . 2 cm .

ANSWER: 2 2 126 cm ; 175 cm

Therefore, the surface area of the pyramid is about 54. 2 255.4 ft . SOLUTION: The pyramid has a slant height of 15 inches and the ANSWER: base is a hexagon with sides of 10.5 inches. 12-4 Volumes2 of Prisms2 and Cylinders 212.1 ft ; 255.4 ft A central angle of the hexagon is or 60°, so the angle formed in the triangle below is 30°.

53. SOLUTION:

The lateral area L of a regular pyramid is ,

where is the slant height and P is the perimeter of Use a trigonometric ratio to find the measure of the the base. apothem a. Here, the base is a square of side 7 cm and the slant height is 9 cm.

Find the lateral area and surface area of the So, the lateral area of the pyramid is 126 cm2. pyramid.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is 472.5 in2.

Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 Therefore, the surface area of the pyramid is about 126 cm ; 175 cm 758.9 in2.

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

Find the lateral area and surface area of the pyramid.

So, the lateral area of the pyramid is 472.5 in2. The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters.

SOLUTION:

Therefore, the surface area of the pyramid is about 758.9 in2.

ANSWER: 472.5 in2; 758.9 in2

SOLUTION: SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

ANSWER: 57. Find the diameter of a circle with an area of 102 square centimeters. 9.3 in. SOLUTION: Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the

lengths of its diagonals, d and d . The diameter of the circle is about 11.4 m. 1 2

ANSWER: d1 = 12 in. and d2 = 7 + 13 = 20 in. 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION:

ANSWER: 2 120 in

ANSWER:

2 378 m

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

Find the volume of each prism.

1. ANSWER: 3 SOLUTION: 396 in The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. 3. the oblique rectangular prism shown at the right

SOLUTION: If two solids have the same height h and the same 3 The volume is 108 cm . cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 108 cm3

ANSWER: 3 396 in ANSWER: 3. the oblique rectangular prism shown at the right 3 218.4 cm

Find the volume of each cylinder. Round to the nearest tenth.

ANSWER: ANSWER: 206.4 ft3 26.95 m3

ANSWER: 1025.4 cm3

ANSWER: 8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches 206.4 ft3 SOLUTION:

7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters

SOLUTION: Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: D ANSWER: 1025.4 cm3 CCSS SENSE-MAKING Find the volume of each prism. 8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches

SOLUTION: 10. SOLUTION: The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

ANSWER: 3 410.1 in

MULTIPLE CHOICE 9. A rectangular lap pool ANSWER: measures 80 feet long by 20 feet wide. If it needs to 3 be filled to four feet deep and each cubic foot holds 30 in 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000 D 48,000 11. SOLUTION: SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the prism is 14 m.

Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: ANSWER: 3 D 539 m CCSS SENSE-MAKING Find the volume of each prism.

12. 10. SOLUTION: SOLUTION: The base is a right triangle with a leg length of 9 cm The base is a rectangle of length 3 in. and width 2 in. and the hypotenuse of length 15 cm. The height of the prism is 5 in.

Use the Pythagorean Theorem to find the height of the base.

ANSWER: 30 in3

The height of the prism is 6 cm.

11. SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the ANSWER: prism is 14 m. 3 324 cm

ANSWER: 13. 3 539 m SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 12.

14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have The height of the prism is 6 cm. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 324 cm ANSWER: 3 2040 cm

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is ANSWER: 1534.25 in3 ANSWER: 58.14 ft3 CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. 14. an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters SOLUTION: 16. If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have r = 5 yd and h = 18 yd the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 1413.7 yd ANSWER: 3 2040 cm

15. a square prism with a base edge of 9.5 inches and a height of 17 inches 17. SOLUTION: SOLUTION: If two solids have the same height h and the same r = 6 cm and h = 3.6 cm. cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 407.2 cm3 ANSWER: 1534.25 in3

CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. 18. SOLUTION: r = 5.5 in.

16. Use the Pythagorean Theorem to find the height of SOLUTION: the cylinder. r = 5 yd and h = 18 yd

ANSWER: 3 1413.7 yd

Now you can find the volume.

17. SOLUTION: r = 6 cm and h = 3.6 cm.

ANSWER: 3 823.0 in

ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a

rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil Now you can find the volume. in the planter if the planter is filled to inches below the top? SOLUTION:

The planter is to be filled inches below the top, so

ANSWER: 3 823.0 in

19. SOLUTION: ANSWER: If two solids have the same height h and the same 3 2740.5 in cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 21. SHIPPING A box 18 centimeters by 9 centimeters an oblique one of the same height and cross sectional by 15 centimeters is being used to ship two

area are same. cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown r = 7.5 mm and h = 15.2 mm. at the right. What is the volume of the empty space in the box?

ANSWER: 2686.1 mm3 SOLUTION: The volume of the empty space is the difference of 20. PLANTER A planter is in the shape of a volumes of the rectangular prism and the cylinders.

rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to inches

below the top? SOLUTION:

The planter is to be filled inches below the top, so ANSWER: 521.5 cm3

22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? SOLUTION: ANSWER: 3 3 2740.5 in V = 10 ft and h = 2 ft Use the formula to find r.

Therefore, the diameter of the cylinders should be about 2.52 ft.

ANSWER: 2.52 ft SOLUTION: Find the volume of the solid formed by each The volume of the empty space is the difference of net. volumes of the rectangular prism and the cylinders.

23.

SOLUTION: The middle piece of the net is the front of the solid. ANSWER: The top and bottom pieces are the bases and the 3 521.5 cm pieces on the ends are the side faces. This is a triangular prism. 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, One leg of the base 14 cm and the hypotenuse 31.4 and 10 cubic feet of sand. To transport the correct cm. Use the Pythagorean Theorem to find the height amount of sand, they want to create cylinders that of the base. are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be?

SOLUTION: The height of the prism is 20 cm.

3 V = 10 ft and h = 2 ft Use the formula to find r. The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism.

ANSWER: Therefore, the diameter of the cylinders should be 3934.9 cm3 about 2.52 ft.

ANSWER: 2.52 ft Find the volume of the solid formed by each net.

The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism.

ANSWER: 3 48.9 m ANSWER: 3934.9 cm3 25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the same radius, what is the height of the larger can?

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m.

ANSWER: 35.1 cm

a. When the height is tripled, h = 3h.

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r.

The height of the new can will be 35.1 cm.

SOLUTION:

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h a. = 5 cm. When the height is tripled, h = 3h.

Compare to the original volume.

When the height is tripled, the volume is multiplied by 3. b. When the radius is tripled, r = 3r. The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27.

d. The volume is multiplied by

So, when the radius is tripled, the volume is multiplied 27. SOIL A soil scientist wants to determine the bulk by 9. density of a potting soil to assess how well a specific

plant will grow in it. The density c. When the height and the radius are tripled, r = 3r of the soil sample is the ratio of its weight to its and h = 3h. volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its When the height and the radius are tripled, the weight in pounds? volume is multiplied by 27. SOLUTION:

d. When the dimensions are exchanged, r = 8 and h = 5 cm.

a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

Compare to the original volume.

The volume is multiplied by .

ANSWER: The weight of the soil is the weight of the pot with soil minus the weight of the pot. a. The volume is multiplied by 3. W = 20 – 5 = 15 lbs. b. The volume is multiplied by 32 or 9. 3 c. The volume is multiplied by 3 or 27. The soil density is thus:

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant plant will grow in it. The density should grow fairly well. of the soil sample is the ratio of its weight to its volume. c.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil’s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? ANSWER: 3 SOLUTION: a. 0.0019 lb / in

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired 3 bulk density of 0.0018 lb / in .

c. 8.3 lb a. First calculate the volume of soil in the pot. Then Find the volume of each composite solid. Round divide the weight of the soil by the volume. to the nearest tenth if necessary.

b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

c. ANSWER: 225 cm3

ANSWER: 3 a. 0.0019 lb / in

29. b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired SOLUTION: 3 bulk density of 0.0018 lb / in . The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid c. 8.3 lb is the sum of the volumes of the two rectangular prisms. Find the volume of each composite solid. Round to the nearest tenth if necessary.

28. SOLUTION: ANSWER: The solid is a combination of two rectangular prisms. 3 The base of one rectangular prism is 5 cm by 3 cm 120 m and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm.

30. SOLUTION: The solid is a combination of a rectangular prism and ANSWER: two half cylinders. 225 cm3

29. ANSWER: 3 SOLUTION: 260.5 in The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid 31. MANUFACTURING A can 12 centimeters tall fits is the sum of the volumes of the two rectangular into a rubberized cylindrical holder that is 11.5 prisms. centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

ANSWER:

Therefore, the volume of the rubberized material is 3 about 304.1 cm .

ANSWER:

3 304.1 cm ANSWER: Find each measure to the nearest tenth. 260.5 in3 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 31. MANUFACTURING A can 12 centimeters tall fits centimeters. What is the height? into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the SOLUTION: thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder?

Therefore, the volume of the rubberized material is about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 The radius is 6. Find the volume. centimeters. What is the height? SOLUTION:

ANSWER: 678.6 in3

SOLUTION: Use the surface area formula to solve for r.

Find the volume.

ANSWER:

3 The radius is 6. Find the volume. 576 in

ANSWER: SOLUTION: 3 The volume will be the highest when the diameter is 678.6 in 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively. 34. A rectangular prism has a surface area of 432

square inches, a height of 6 inches, and a width of 12 Find the difference between the volumes. inches. What is the volume?

SOLUTION: Use the surface area formula to find the length of the base of the prism.

ANSWER:

ANSWER: 3 576 in SOLUTION: 35. ARCHITECTURE A cylindrical stainless steel The swimming pool is a combination of a rectangular column is used to hide a ventilation system in a new prism and a trapezoidal prism. The base of the building. According to the specifications, the diameter rectangular prism is 6 ft by 10 ft and the height is 15 of the column can be between 30 centimeters and 95 ft. The bases of the trapezoidal prism are 6 ft and 3 centimeters. The height is to be 500 centimeters. ft long and the height of the base is 10 ft. The height What is the difference in volume between the largest of the trapezoidal prism is 15 ft. The total volume of and smallest possible column? Round to the nearest the solid is the sum of the volumes of the two tenth cubic centimeter. prisms. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes.

ANSWER: 3 1575 ft

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool. SOLUTION: Find the volume of the original container.

ANSWER:

ANSWER: 38. DESIGN Sketch and label (in inches) three different 3 designs for a dry ingredient measuring cup that holds 1575 ft 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) 37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy SOLUTION: milk in each container will be increased by 25%. The Sample answers: company wants the base of the container to stay the same. What will be the height of the new containers? For any cylindrical container, we have the following equation for volume:

The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h.

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each 3 drawing. (1 cup ≈ 14.4375 in )

SOLUTION: For any rectangular container, the volume equation is: Sample answers:

For any cylindrical container, we have the following equation for volume: Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER:

For any rectangular container, the volume equation is: Sample answers:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

SOLUTION:

The base of the prism can be divided into 5 ANSWER: congruent triangles of a base 8 cm and the Sample answers: corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism.

Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd c. ANALYTICAL Describe which change affects 12 feet = 4 yd the volume of the cylinder more: multiplying the 4 in. = yd height by x or multiplying the radius by x. Explain. SOLUTION: Find the volume. a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky.

b. Find the volume of each. The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option.

ANSWER: Because 2.96 yd3 of concrete are needed, the second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders The volume of the square prism is greater.

. c. Do each scenario. a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters.

b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain.

c. ANALYTICAL Describe which change affects the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain.

SOLUTION: Assuming x > 1, multiplying the radius by x makes 2 a. The oblique cylinder should look like the right the volume x times greater. cylinder (same height and size), except that it is 2 pushed a little to the side, like a slinky. For example, if x = 0.5, then x = 0.25, which is less than x.

ANSWER: a.

b. Find the volume of each.

c. Multiplying the radius by x ; since the volume is

Assuming x > 1, multiplying the radius by x makes SOLUTION: 2 the volume x times greater. Francisco; Valerie incorrectly used as the

length of one side of the triangular base. Francisco 2 For example, if x = 0.5, then x = 0.25, which is less used a different approach, but his solution is correct. than x. Francisco used the standard formula for the volume ANSWER: of a solid, V = Bh. The area of the base, B, is one- a. half the apothem multiplied by the perimeter of the base.

ANSWER:

Francisco; Valerie incorrectly used as the b. Greater than; a square with a side length of 6 m length of one side of the triangular base. Francisco 2 has an area of 36 m . A circle with a diameter of 6 used a different approach, but his solution is correct. m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is 43. CHALLENGE A cylindrical can is used to fill a greater. container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes. c. Multiplying the radius by x ; since the volume is

represented by π r 2 h, multiplying the height by x a. makes the volume x times greater. Multiplying the rectangular prism

radius by x makes the volume x 2 times greater, b. assuming x > 1. square prism

SOLUTION:

3 The volume of the can is 20π in . It takes three full SOLUTION: cans to fill the container, so the volume of the Francisco; Valerie incorrectly used as the container is 60π in 3. length of one side of the triangular base. Francisco used a different approach, but his solution is correct. a. Choose some basic values for 2 of the sides, and

then determine the third side. Base: 3 by 5. Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the base.

3 by 5 by 4π ANSWER: b. Francisco; Valerie incorrectly used as the Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

a. rectangular prism

b. square prism 5 by 5 by

c. triangular prism with a right triangle as the base c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4.

SOLUTION:

3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the 3 by 4 by 10π 3 container is 60π in . ANSWER:

Sample answers: a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. a. 3 by 5 by 4π

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 10π in.

3 by 5 by 4π 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet b. Choose some basic values for 2 of the sides, and gardening forum. then determine the third side. Base: 5 by 5. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. 5 by 5 by ANSWER: c. Choose some basic values for 2 of the sides, and Sample answer: The nursery means a cubic yard, 3 then determine the third side. Base: Legs: 3 by 4. which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest

3 by 4 by 10π values to choose. ANSWER: Sample answer: Sample answers:

a. 3 by 5 by 4π

b. 5 by 5 by

ANSWER: c. base with legs measuring 3 in. and 4 in., height 10π in. Sample answer:

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum.

I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I 46. REASONING Determine whether the following know that a yard is 3 feet, but what is a yard of statement is true or false. Explain. soil? How do I know what to order? Two cylinders with the same height and the same lateral area must have the same volume. SOLUTION: Sample answer: The nursery means a cubic yard, SOLUTION: 3 which is 3 or 27 cubic feet. Find the volume of your True; if two cylinders have the same height (h1 = h2) garden in cubic feet and divide by 27 to determine and the same lateral area (L = L ), the circular the number of cubic yards of soil needed. 1 2 bases must have the same area. ANSWER: Sample answer: The nursery means a cubic yard, which is 33 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters.

47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different?

The radii must also be equal.

ANSWER: True; if two cylinders have the same height and the same lateral area, the circular bases must have the 2 same area. Therefore, πr h is the same for each cylinder.

47. WRITING IN MATH How are the formulas for ANSWER: the volume of a prism and the volume of a cylinder B similar? How are they different? 49. A cylindrical tank used for oil storage has a height SOLUTION: that is half the length of its radius. If the volume of Both formulas involve multiplying the area of the the tank is 1,122,360 ft3, what is the tank’s radius? base by the height. The base of a prism is a polygon, F 89.4 ft so the expression representing the area varies, G 178.8 ft depending on the type of polygon it is. The base of a H 280.9 ft 2 cylinder is a circle, so its area is πr . J 561.8 ft ANSWER: SOLUTION: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr .

D 11.5 cm SOLUTION: ANSWER: F 50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square?

Therefore, the correct choice is C.

ANSWER: C ANSWER: Find the lateral area and surface area of each F regular pyramid. Round to the nearest tenth if 50. SHORT RESPONSE What is the ratio of the area necessary. of the circle to the area of the square?

52.

The slant height is the height of each of the ANSWER: congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C

52. SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height.

The perimeter is P = 3 × 10 or 30 feet.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. So, the lateral area of the pyramid is about 212.1 ft2.

Therefore, the surface area of the pyramid is about 2 255.4 ft . ANSWER: 212.1 ft2; 255.4 ft2

The perimeter is P = 3 × 10 or 30 feet.

53. SOLUTION:

The lateral area L of a regular pyramid is ,

So, the lateral area of the pyramid is 126 cm2.

So, the lateral area of the pyramid is about 212.1 ft2. The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

Therefore, the surface area of the pyramid is about 2 255.4 ft . Therefore, the surface area of the pyramid is 175 2 ANSWER: cm . 2 2 212.1 ft ; 255.4 ft ANSWER: 2 2 126 cm ; 175 cm

53. 54. SOLUTION: SOLUTION: The pyramid has a slant height of 15 inches and the The lateral area L of a regular pyramid is , base is a hexagon with sides of 10.5 inches. where is the slant height and P is the perimeter of A central angle of the hexagon is or 60°, so the the base. angle formed in the triangle below is 30°. Here, the base is a square of side 7 cm and the slant height is 9 cm.

So, the lateral area of the pyramid is 126 cm2.

Use a trigonometric ratio to find the measure of the The surface area S of a regular apothem a. pyramid is , where L is the lateral area and B is the area of the base.

Find the lateral area and surface area of the pyramid. Therefore, the surface area of the pyramid is 175 cm2.

ANSWER: 2 2 126 cm ; 175 cm

So, the lateral area of the pyramid is 472.5 in2.

ANSWER: 472.5 in2; 758.9 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the So, the lateral area of the pyramid is 472.5 in2. diameter. SOLUTION:

Therefore, the surface area of the pyramid is about 758.9 in2. The diameter of the circle is about 8.3 m.

ANSWER: ANSWER: 12-4 Volumes of Prisms and Cylinders 8.3 m 472.5 in2; 758.9 in2

SOLUTION: The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 +

2 9)(2) + 13(9) = 205 in . The diameter of the circle is about 11.4 m.

ANSWER: ANSWER: 205 in2 11.4 cm Find the indicated measure. Round to the 58. The area of a circle is 191 square feet. Find the nearest tenth. radius. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION: SOLUTION:

ANSWER: eSolutions Manual - Powered by Cognero 9.3 in. Page 22 The diameter of the circle is about 11.4 m. Find the area of each trapezoid, rhombus, or ANSWER: kite. 11.4 cm 58. The area of a circle is 191 square feet. Find the radius. SOLUTION: 60. SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square

inches. ANSWER: 2 SOLUTION: 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of

the height h and the sum of the lengths of its bases,

b1 and b2. ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

ANSWER: 2 60. 378 m SOLUTION: The area A of a kite is one half the product of the lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 2 378 m Find the volume of each prism.

1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

3 The volume is 108 cm .

ANSWER: 108 cm3

2. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

Find the volume of each prism. ANSWER: 3 396 in

3. the oblique rectangular prism shown at the right 1. SOLUTION: The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

3 The volume is 108 cm .

ANSWER: 108 cm3

ANSWER: 26.95 m3

4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters 2. SOLUTION: SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have The volume V of a prism is V = Bh, where B is the the same volume. So, the volume of a right prism and

area of a base and h is the height of the prism. an oblique one of the same height and cross sectional

area are same.

ANSWER:

3 218.4 cm ANSWER: 3 Find the volume of each cylinder. Round to the 396 in nearest tenth. 3. the oblique rectangular prism shown at the right

ANSWER: 206.4 ft3

ANSWER: 3 26.95 m 6. SOLUTION: 4. an oblique pentagonal prism with a base area of 42 If two solids have the same height h and the same square centimeters and a height of 5.2 centimeters cross-sectional area B at every level, then they have SOLUTION: the same volume. So, the volume of a right cylinder If two solids have the same height h and the same and an oblique one of the same height and cross cross-sectional area B at every level, then they have sectional area are same. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

ANSWER: 3 1357.2 m ANSWER: 3 7. a cylinder with a diameter of 16 centimeters and a 218.4 cm height of 5.1 centimeters Find the volume of each cylinder. Round to the SOLUTION: nearest tenth.

5. SOLUTION: ANSWER: 1025.4 cm3

8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches SOLUTION:

ANSWER: 206.4 ft3

ANSWER: 3 410.1 in 6. 9. MULTIPLE CHOICE A rectangular lap pool SOLUTION: measures 80 feet long by 20 feet wide. If it needs to If two solids have the same height h and the same be filled to four feet deep and each cubic foot holds cross-sectional area B at every level, then they have 7.5 gallons, how many gallons will it take to fill the the same volume. So, the volume of a right cylinder lap pool? and an oblique one of the same height and cross A 4000 sectional area are same. B 6400 C 30,000 D 48,000 SOLUTION:

ANSWER: 3 Each cubic foot holds 7.5 gallons of water. So, the 1357.2 m amount of water required to fill the pool is 6400(7.5) = 48,000. 7. a cylinder with a diameter of 16 centimeters and a

height of 5.1 centimeters Therefore, the correct choice is D. SOLUTION: ANSWER: D CCSS SENSE-MAKING Find the volume of each prism.

ANSWER: 3 10. 1025.4 cm SOLUTION: 8. a cylinder with a radius of 4.2 inches and a height of The base is a rectangle of length 3 in. and width 2 in. 7.4 inches The height of the prism is 5 in. SOLUTION:

ANSWER: ANSWER: 30 in3 3 410.1 in

9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the 11. lap pool? A 4000 SOLUTION: B 6400 The base is a triangle with a base length of 11 m and C 30,000 the corresponding height of 7 m. The height of the D 48,000 prism is 14 m.

SOLUTION:

Each cubic foot holds 7.5 gallons of water. So, the ANSWER: amount of water required to fill the pool is 6400(7.5) 3 = 48,000. 539 m

Therefore, the correct choice is D.

ANSWER: D 12. CCSS SENSE-MAKING Find the volume of each prism. SOLUTION: The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm.

10. Use the Pythagorean Theorem to find the height of the base. SOLUTION:

The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in.

ANSWER: 3 30 in The height of the prism is 6 cm.

ANSWER: 3 11. 324 cm SOLUTION: The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the prism is 14 m.

13.

SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have ANSWER: the same volume. So, the volume of a right prism and 3 an oblique one of the same height and cross sectional 539 m area are same.

The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism.

2 12. B = 11.4 ft and h = 5.1 ft. Therefore, the volume is

SOLUTION: The base is a right triangle with a leg length of 9 cm ANSWER: and the hypotenuse of length 15 cm. 58.14 ft3

The height of the prism is 6 cm.

ANSWER:

3 2040 cm ANSWER: 324 cm3 15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional 13. area are same. SOLUTION: If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. ANSWER: The volume V of a prism is V = Bh, where B is the 3 area of a base and h is the height of the prism. 1534.25 in

CCSS SENSE-MAKING Find the volume of 2 B = 11.4 ft and h = 5.1 ft. Therefore, the volume is each cylinder. Round to the nearest tenth.

ANSWER: 3 58.14 ft 16.

17. ANSWER: 3 SOLUTION: 2040 cm r = 6 cm and h = 3.6 cm. 15. a square prism with a base edge of 9.5 inches and a height of 17 inches SOLUTION: If two solids have the same height h and the same

cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional ANSWER: area are same. 3 407.2 cm

18. ANSWER: 3 SOLUTION: 1534.25 in r = 5.5 in. CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. Use the Pythagorean Theorem to find the height of the cylinder.

16. SOLUTION: r = 5 yd and h = 18 yd

ANSWER: Now you can find the volume. 3 1413.7 yd

17. ANSWER: SOLUTION: 3 r = 6 cm and h = 3.6 cm. 823.0 in

19.

SOLUTION: ANSWER: If two solids have the same height h and the same cross-sectional area B at every level, then they have 407.2 cm3 the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same.

r = 7.5 mm and h = 15.2 mm.

18. SOLUTION: r = 5.5 in.

Use the Pythagorean Theorem to find the height of the cylinder. ANSWER: 2686.1 mm3

SOLUTION: Now you can find the volume. The planter is to be filled inches below the top, so

ANSWER: 3 823.0 in

ANSWER: 3 2740.5 in

r = 7.5 mm and h = 15.2 mm.

SOLUTION: The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders.

ANSWER: 2686.1 mm3

20. PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep,

and 12 inches high. What is the volume of potting soil ANSWER: in the planter if the planter is filled to inches 521.5 cm3 below the top? SANDCASTLES SOLUTION: 22. In a sandcastle competition, contestants are allowed to use only water, shovels, The planter is to be filled inches below the top, so and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be?

SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3 2740.5 in

One leg of the base 14 cm and the hypotenuse 31.4 ANSWER: cm. Use the Pythagorean Theorem to find the height 521.5 cm3 of the base.

SOLUTION:

3 V = 10 ft and h = 2 ft Use the formula to find r.

ANSWER: 3934.9 cm3

Therefore, the diameter of the cylinders should be about 2.52 ft.

23. The radius is 1.8 m, the height is 4.8 m, and the slant SOLUTION: height is 6 m. The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the If two solids have the same height h and the same pieces on the ends are the side faces. This is a cross-sectional area B at every level, then they have triangular prism. the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base.

The height of the prism is 20 cm.

The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism. ANSWER: 3 48.9 m

25. FOOD A cylindrical can of baked potato chips has a height of 27 centimeters and a radius of 4 centimeters. A new can is advertised as being 30% larger than the regular can. If both cans have the ANSWER: same radius, what is the height of the larger can? 3934.9 cm3

SOLUTION: The volume of the smaller can is

The volume of the new can is 130% of the smaller can, with the same radius. 24.

SOLUTION: The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique.

The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m. The height of the new can will be 35.1 cm. If two solids have the same height h and the same ANSWER: cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and 35.1 cm an oblique one of the same height and cross sectional 26. CHANGING DIMENSIONS A cylinder has a area are same. radius of 5 centimeters and a height of 8 centimeters.

Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. d. The dimensions are exchanged.

SOLUTION: ANSWER: 3 48.9 m

same radius, what is the height of the larger can?

So, when the radius is tripled, the volume is multiplied by 9.

The height of the new can will be 35.1 cm. c. When the height and the radius are tripled, r = 3r ANSWER: and h = 3h. 35.1 cm

26. CHANGING DIMENSIONS A cylinder has a radius of 5 centimeters and a height of 8 centimeters. Describe how each change affects the volume of the cylinder. a. The height is tripled. b. The radius is tripled. c. Both the radius and the height are tripled. When the height and the radius are tripled, the d. The dimensions are exchanged. volume is multiplied by 27.

SOLUTION: d. When the dimensions are exchanged, r = 8 and h = 5 cm.

a. When the height is tripled, h = 3h. Compare to the original volume.

The volume is multiplied by .

ANSWER: When the height is tripled, the volume is multiplied by a. The volume is multiplied by 3. 3. b. The volume is multiplied by 32 or 9. b. When the radius is tripled, r = 3r. c. 3 The volume is multiplied by 3 or 27.

d. The volume is multiplied by

27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume. So, when the radius is tripled, the volume is multiplied by 9. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 c. When the height and the radius are tripled, r = 3r pounds, what is the soil’s bulk density? b. Assuming and h = 3h. that all other factors are favorable, how well should a plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? SOLUTION:

When the height and the radius are tripled, the volume is multiplied by 27.

d. When the dimensions are exchanged, r = 8 and h = 5 cm. a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume.

Compare to the original volume. The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The volume is multiplied by . The soil density is thus: ANSWER: a. The volume is multiplied by 3. 2 b. The volume is multiplied by 3 or 9. 3 c. The volume is multiplied by 3 or 27. b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant

d. The volume is multiplied by should grow fairly well.

c. 27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific

plant will grow in it. The density of the soil sample is the ratio of its weight to its volume.

a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 ANSWER: pounds, what is the soil’s bulk density? b. Assuming 3 that all other factors are favorable, how well should a a. 0.0019 lb / in plant grow in this soil if a bulk density of 0.018 pound per square inch is desirable for root growth? Explain. b. The plant should grow well in this soil since the c. If a bag of this soil holds 2.5 cubic feet, what is its bulk density of 0.0019 lb / in3 is close to the desired weight in pounds? 3 bulk density of 0.0018 lb / in . SOLUTION: c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

and the height is 11 cm. The base of the other prism The weight of the soil is the weight of the pot with is 4 cm by 3 cm and the height is 5 cm. soil minus the weight of the pot. W = 20 – 5 = 15 lbs.

The soil density is thus:

ANSWER:

225 cm3 b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well.

29.

c. 8.3 lb

Find the volume of each composite solid. Round to the nearest tenth if necessary.

ANSWER: 3 120 m 28. SOLUTION: The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism 30. is 4 cm by 3 cm and the height is 5 cm. SOLUTION: The solid is a combination of a rectangular prism and two half cylinders.

ANSWER: 225 cm3

ANSWER: 260.5 in3

31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 29. centimeters tall, including 1 centimeter for the SOLUTION: thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the The solid is a combination of a rectangular prism and volume of the rubberized material that makes up the a right triangular prism. The total volume of the solid holder? is the sum of the volumes of the two rectangular prisms.

SOLUTION:

The volume of the rubberized material is the difference between the volumes of the container and ANSWER: the space used for the can. The container has a 3 120 m radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

30. SOLUTION:

The solid is a combination of a rectangular prism and two half cylinders. Therefore, the volume of the rubberized material is

about 304.1 cm3.

ANSWER: 3 304.1 cm

Find each measure to the nearest tenth.

32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9

centimeters. What is the height? ANSWER: SOLUTION: 260.5 in3

ANSWER: 5.7 cm

the space used for the can. The container has a radius of and a height of 11.5 cm. The empty space used to keep the can has a radius of 3.25 cm and a height of 11.5 – 1 = 10.5 cm. The 2 volume V of a cylinder is V = πr h, where r is the radius of the base and h is the height of the cylinder.

Therefore, the volume of the rubberized material is 3 about 304.1 cm .

ANSWER: The radius is 6. Find the volume. 3 304.1 cm

Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height?

SOLUTION:

ANSWER: 678.6 in3

5.7 cm

33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? SOLUTION: Use the surface area formula to solve for r.

Find the volume.

ANSWER: 3 576 in

35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter

of the column can be between 30 centimeters and 95

The radius is 6. Find the volume. centimeters. The height is to be 500 centimeters.

What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is

when the radii are 47.5 cm and 15 cm respectively.

Find the difference between the volumes. ANSWER:

678.6 in3

ANSWER: 3,190,680.0 cm3

36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown

in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool. Find the volume.

SOLUTION:

The swimming pool is a combination of a rectangular ANSWER: prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 3 576 in ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height 35. ARCHITECTURE A cylindrical stainless steel of the trapezoidal prism is 15 ft. The total volume of column is used to hide a ventilation system in a new the solid is the sum of the volumes of the two building. According to the specifications, the diameter prisms. of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. SOLUTION: The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.That is when the radii are 47.5 cm and 15 cm respectively. ANSWER: 3 Find the difference between the volumes. 1575 ft

ANSWER: 3,190,680.0 cm3

CCSS MODELING 36. The base of a rectangular SOLUTION: swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown Find the volume of the original container.

in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool.

of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms. ANSWER:

38. DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3)

The volume of the new container is 125% of the original container, with the same base dimensions. If we choose a height of say 4 in., then we can solve Use 1.25V and B to find h. for the radius.

ANSWER:

38. DESIGN Sketch and label (in inches) three different For any rectangular container, the volume equation is: designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup ≈ 14.4375 in3) Choose numbers for any two of the dimensions and SOLUTION: we can solve for the third. Let l = 2.25 in. and w = Sample answers: 2.5 in.

For any cylindrical container, we have the following equation for volume:

If we choose a height of say 4 in., then we can solve for the radius.

ANSWER: Sample answers:

For any rectangular container, the volume equation is:

Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

ANSWER: Sample answers: Therefore, the volume of the pentagonal prism is

ANSWER: 3 1100 cm ; Each triangular prism has a base area of 2 or 22 cm and a height of 10 cm.

40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid $2225 for the project. A second contractor bid $500 per cubic yard for the new patio and $700 for removal of the old patio. Which is the less expensive option? Explain. SOLUTION: Convert all of the dimensions to yards.

20 feet = yd 12 feet = 4 yd 4 in. = yd

Find the volume.

39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism.

SOLUTION: The total cost for the second contractor is about The base of the prism can be divided into 5 . congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal Therefore, the second contractor is a less expensive prism is a combination of 5 triangular prisms of height option. 10 cm. Find the base area of each triangular prism. ANSWER: Because 2.96 yd3 of concrete are needed, the Therefore, the volume of the pentagonal prism is second contractor is less expensive at $2181.50.

41. MULTIPLE REPRESENTATIONS In this ANSWER: problem, you will investigate right and oblique cylinders 3 1100 cm ; Each triangular prism has a base area of . 2 a. GEOMETRIC Draw a right cylinder and an or 22 cm and a height of 10 cm. oblique cylinder with a height of 10 meters and a diameter of 6 meters. 40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 b. VERBAL A square prism has a height of 10 feet long, 12 feet wide, and 4 inches thick. One meters and a base edge of 6 meters. Is its volume contractor bid $2225 for the project. A second greater than, less than, or equal to the volume of contractor bid $500 per cubic yard for the new patio the cylinder? Explain. and $700 for removal of the old patio. Which is the

less expensive option? Explain. c. ANALYTICAL Describe which change affects SOLUTION: the volume of the cylinder more: multiplying the Convert all of the dimensions to yards. height by x or multiplying the radius by x. Explain. SOLUTION: 20 feet = yd a. The oblique cylinder should look like the right 12 feet = 4 yd cylinder (same height and size), except that it is

4 in. = yd pushed a little to the side, like a slinky.

Find the volume.

b. Find the volume of each.

The total cost for the second contractor is about .

Therefore, the second contractor is a less expensive option.

ANSWER:

Because 2.96 yd3 of concrete are needed, the The volume of the square prism is greater. second contractor is less expensive at $2181.50. c. Do each scenario.

2 c. ANALYTICAL Describe which change affects the volume x times greater. the volume of the cylinder more: multiplying the 2 height by x or multiplying the radius by x. Explain. For example, if x = 0.5, then x = 0.25, which is less than x. SOLUTION: a. The oblique cylinder should look like the right ANSWER: cylinder (same height and size), except that it is a. pushed a little to the side, like a slinky.

The volume of the square prism is greater. 42. CCSS CRITIQUE Franciso and Valerie each c. Do each scenario. calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning.

SOLUTION: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Assuming x > 1, multiplying the radius by x makes 2 Francisco used the standard formula for the volume the volume x times greater. of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the For example, if x = 0.5, then x2 = 0.25, which is less base. than x.

ANSWER: a. ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

m has an area of 9π or 28.3 m2. Since the heights are the same, the volume of the square prism is a. rectangular prism greater. b. square prism c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x c. triangular prism with a right triangle as the base makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > 1.

42. CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 SOLUTION: units. Is either of them correct? Explain your 3 reasoning. The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the container is 60π in 3.

a. Choose some basic values for 2 of the sides, and SOLUTION: then determine the third side. Base: 3 by 5.

Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is one- half the apothem multiplied by the perimeter of the 3 by 5 by 4π

base.

b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5.

ANSWER: Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct.

43. CHALLENGE A cylindrical can is used to fill a container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the

container if it is each of the following shapes. 5 by 5 by

c. a. rectangular prism Choose some basic values for 2 of the sides, and

then determine the third side. Base: Legs: 3 by 4. b. square prism

c. triangular prism with a right triangle as the base

3 by 4 by 10π SOLUTION: ANSWER: Sample answers: 3 The volume of the can is 20π in . It takes three full cans to fill the container, so the volume of the a. 3 by 5 by 4π

container is 60π in 3. b. 5 by 5 by a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum.

then determine the third side. Base: 5 by 5. SOLUTION: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine 5 by 5 by the number of cubic yards of soil needed.

OPEN ENDED c. Choose some basic values for 2 of the sides, and 45. Draw and label a prism that has a

then determine the third side. Base: Legs: 3 by 4. volume of 50 cubic centimeters. SOLUTION: Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest values to choose.

Sample answer:

3 by 4 by 10π

ANSWER: Sample answers: ANSWER: a. π 3 by 5 by 4 Sample answer:

b. 5 by 5 by

c. base with legs measuring 3 in. and 4 in., height 10π in.

44. WRITING IN MATH Write a helpful response to 46. REASONING Determine whether the following the following question posted on an Internet statement is true or false. Explain. gardening forum. Two cylinders with the same height and the same lateral area must have the same volume. I am new to gardening. The nursery will deliver a SOLUTION: truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of True; if two cylinders have the same height (h1 = h2) soil? How do I know what to order? and the same lateral area (L1 = L2), the circular SOLUTION: bases must have the same area. Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed.

ANSWER: Sample answer: The nursery means a cubic yard, 3 which is 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine The radii must also be equal. the number of cubic yards of soil needed. ANSWER: 45. OPEN ENDED Draw and label a prism that has a True; if two cylinders have the same height and the volume of 50 cubic centimeters. same lateral area, the circular bases must have the 2 SOLUTION: same area. Therefore, πr h is the same for each Choose 3 values that multiply to make 50. The cylinder. factors of 50 are 2, 5, 5, so these are the simplest WRITING IN MATH values to choose. 47. How are the formulas for the volume of a prism and the volume of a cylinder

Sample answer: similar? How are they different? SOLUTION: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2. ANSWER: Sample answer: ANSWER: Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The 2 base of a cylinder is a circle, so its area is πr . 46. REASONING Determine whether the following 48. The volume of a triangular prism is 1380 cubic statement is true or false. Explain. centimeters. Its base is a right triangle with legs Two cylinders with the same height and the same measuring 8 centimeters and 15 centimeters. What is lateral area must have the same volume. the height of the prism? A 34.5 cm SOLUTION: B 23 cm True; if two cylinders have the same height (h1 = h2) C 17 cm and the same lateral area (L1 = L2), the circular D 11.5 cm bases must have the same area. SOLUTION:

The radii must also be equal.

ANSWER: True; if two cylinders have the same height and the ANSWER: same lateral area, the circular bases must have the B 2 same area. Therefore, πr h is the same for each 49. A cylindrical tank used for oil storage has a height cylinder. that is half the length of its radius. If the volume of 3 47. WRITING IN MATH How are the formulas for the tank is 1,122,360 ft , what is the tank’s radius? the volume of a prism and the volume of a cylinder F 89.4 ft similar? How are they different? G 178.8 ft H 280.9 ft SOLUTION: J 561.8 ft Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, SOLUTION: so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr2.

SOLUTION: The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown.

ANSWER: ANSWER: B

49. A cylindrical tank used for oil storage has a height 51. SAT/ACT A county proposes to enact a new 0.5% that is half the length of its radius. If the volume of property tax. What would be the additional tax 3 the tank is 1,122,360 ft , what is the tank’s radius? amount for a landowner whose property has a F 89.4 ft taxable value of $85,000? G 178.8 ft A $4.25 H 280.9 ft B $170 J 561.8 ft C $425 D $4250 SOLUTION: E $42,500 SOLUTION: Find the 0.5% of $85,000.

Therefore, the correct choice is C.

ANSWER: C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

ANSWER: F 52. 50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square? SOLUTION:

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. SOLUTION: The radius of the circle is 2x and the length of each The slant height is the height of each of the side of the square is 4x. So, the ratio of the areas can congruent lateral triangular faces. Use the be written as shown. Pythagorean Theorem to find the slant height.

ANSWER:

51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of $85,000? A $4.25

B $170 C $425 D $4250 E $42,500 SOLUTION: Find the 0.5% of $85,000.

Find the perimeter and area of the equilateral Therefore, the correct choice is C. triangle for the base. Use the Pythagorean Theorem ANSWER: to find the height h of the triangle. C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

52.

SOLUTION: The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base.

The slant height is the height of each of the congruent lateral triangular faces. Use the The perimeter is P = 3 × 10 or 30 feet. Pythagorean Theorem to find the slant height.

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid.

So, the lateral area of the pyramid is about 212.1 ft2.

Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle.

Therefore, the surface area of the pyramid is about 2 255.4 ft .

ANSWER: 212.1 ft2; 255.4 ft2

So, the area of the base B is ft2.

Find the lateral area L and surface area S of the regular pyramid. 2 So, the lateral area of the pyramid is 126 cm .

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base.

So, the lateral area of the pyramid is about 212.1 ft2.

Therefore, the surface area of the pyramid is 175 2 cm .

ANSWER: Therefore, the surface area of the pyramid is about 2 2 2 255.4 ft . 126 cm ; 175 cm

ANSWER: 212.1 ft2; 255.4 ft2

54. SOLUTION: The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. 53. A central angle of the hexagon is or 60°, so the

SOLUTION: angle formed in the triangle below is 30°.

The lateral area L of a regular pyramid is , where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm.

Use a trigonometric ratio to find the measure of the apothem a.

So, the lateral area of the pyramid is 126 cm2.

The surface area S of a regular pyramid is , where L is the lateral area and B is the area of the base. Find the lateral area and surface area of the pyramid.

Therefore, the surface area of the pyramid is 175 2 cm2. So, the lateral area of the pyramid is 472.5 in .

ANSWER: 2 2 126 cm ; 175 cm

54. Therefore, the surface area of the pyramid is about SOLUTION: 758.9 in2. The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60°, so the ANSWER: angle formed in the triangle below is 30°. 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

Use a trigonometric ratio to find the measure of the SOLUTION: apothem a. The area that needs to be coated is the sum of the

lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in .

ANSWER: 205 in2

Find the lateral area and surface area of the Find the indicated measure. Round to the pyramid. nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

So, the lateral area of the pyramid is 472.5 in2.

The diameter of the circle is about 8.3 m.

ANSWER: Therefore, the surface area of the pyramid is about 8.3 m 758.9 in2. 57. Find the diameter of a circle with an area of 102 square centimeters. ANSWER: SOLUTION: 472.5 in2; 758.9 in2

55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated?

The diameter of the circle is about 11.4 m. SOLUTION: The area that needs to be coated is the sum of the ANSWER: lateral area and one base area. 11.4 cm Therefore, the area that needs to be coated is 2(13 + 2 9)(2) + 13(9) = 205 in . 58. The area of a circle is 191 square feet. Find the radius. ANSWER: SOLUTION: 205 in2

Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. SOLUTION:

ANSWER: 7.8 ft 59. Find the radius of a circle with an area of 271 square inches.

SOLUTION: The diameter of the circle is about 8.3 m.

ANSWER: 8.3 m

57. Find the diameter of a circle with an area of 102 square centimeters.

SOLUTION:

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or kite.

The diameter of the circle is about 11.4 m.

ANSWER: 11.4 cm 60. 58. The area of a circle is 191 square feet. Find the SOLUTION: radius. The area A of a kite is one half the product of the lengths of its diagonals, d and d . SOLUTION: 1 2

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 12-4ANSWER: Volumes of Prisms and Cylinders 2 120 in 7.8 ft 59. Find the radius of a circle with an area of 271 square inches. SOLUTION: 61. SOLUTION: The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b1 and b2.

ANSWER: 9.3 in. Find the area of each trapezoid, rhombus, or ANSWER: kite. 2 378 m

62. 60. SOLUTION: SOLUTION: d = 2(22) = 44 and d = 2(23) = 46 The area A of a kite is one half the product of the 1 2

lengths of its diagonals, d1 and d2.

d1 = 12 in. and d2 = 7 + 13 = 20 in.

ANSWER: 2 1012 ft

ANSWER: 2 120 in

61. SOLUTION: The area A of a trapezoid is one half the product of eSolutionsthe heightManual h- Poweredand theby sumCognero of the lengths of its bases, Page 23 b1 and b2.

ANSWER: 2 378 m