11-2 Volumes of Prisms and

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

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.

The volume is 108 cm3.

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

SOLUTION: If two solids have the same height h and the same 2. 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.

ANSWER: 218.4 cm3 ANSWER: 396 in3

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Find the volume of each . 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: SOLUTION: 410.1 in3 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: 1357.2 m3

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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? 11. 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:

ANSWER: Each cubic foot holds 7.5 gallons of water. So, the 539 m3 amount of water required to fill the pool is 6400(7.5) = 48,000.

Therefore, the correct choice is D.

ANSWER: D

STRUCTURE 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.

ANSWER: 30 in3

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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.

B = 11.4 ft2 and h = 5.1 ft. Therefore, the volume is

ANSWER: 3 The height of the prism is 6 cm. 58.14 ft

14. an oblique 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 324 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.

ANSWER: 2040 cm3

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

SOLUTION: 17. If two solids have the same height h and the same SOLUTION: cross-sectional area B at every level, then they have r = 6 cm and h = 3.6 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: 407.2 cm3

ANSWER: 1534.25 in3

STRUCTURE Find the volume of each cylinder. 18. Round to the nearest tenth. SOLUTION: r = 5.5 in.

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

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

ANSWER: 823.0 in3

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21. SHIPPING The box shown is being used to ship two cylindrical candles. 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 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: The volume of the empty space is 521.5 cm³. 2686.1 mm3 ANSWER: 20. PLANTER A planter is in the shape of a rectangular 521.5 cm3 prism 18 inches long, inches deep, and 12 inches 22. CHANGING DIMENSIONS A cylinder has a high. What is the volume of potting soil in the planter radius of 5 centimeters and a height of 8 centimeters. if the planter is filled to inches below the top? Describe how each change affects the volume of the cylinder. SOLUTION: a. The height is tripled. b. The radius is tripled. The planter is to be filled inches below the top, so c. Both the radius and the height are tripled. d. The dimensions are exchanged.

SOLUTION:

ANSWER: 2740.5 in3 a. When the height is tripled, h = 3h.

When the height is tripled, the volume is multiplied by eSolutions Manual - Powered by Cognero Page 6 11-2 Volumes of Prisms and Cylinders

3. 23. INSULATION The insulated cup holds 16 ounces of liquid. Find the volume of the insulating material, b. When the radius is tripled, r = 3r. rounded to the nearest cubic inch.

So, when the radius is tripled, the volume is multiplied by 9. SOLUTION: c. When the height and the radius are tripled, r = 3r The volume of the insulated material is the difference and h = 3h. between the volumes of the interior cylinder (which holds the liquid) and the entire cylinder (cup). The inner cylinder has a volume of 16 ounces (which converts to 28.875 cubic inches). Use this to find the radius of the inner cylinder. Note that the height of the inner cylinder is , due to the extra 0.5 insulation at the bottom of the cup. 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.

Therefore, the radius of the inner cylinder is inches, making the entire cup's radius to be inches.

Compare to the original volume. Find the volume of the entire cup.

The volume is multiplied by .

ANSWER: a. The volume is multiplied by 3. The volume of the insulating material is the difference b. The volume is multiplied by 32 or 9. between the volume of the inner cylinder and the c. The volume is multiplied by 33 or 27. volume of the entire cup. d. The volume is multiplied by

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25. CHANGING DIMENSIONS A cereal company wants to increase the volume of each rectangular prism container by 25% without changing the base. Find the height of the new container if the original Therefore, the volume of the insulated material is had a base of 8 inches by 2 inches and a height of about 31 in3. 12 inches. What would the height be if the surface area of the container increased by 25%? ANSWER: SOLUTION: 31 in3 Find the volume of the original container. 24. MODELING The base of a rectangular paint tray is sloped as shown below. Find the volume of paint it takes to fill the tray. 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.

Next, find the surface area of the original container. SOLUTION: The paint tray is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 8.9 cm by 45 cm and the height is 8.4 cm. The surface area of the new container is 125% of the The bases of the trapezoidal prism are 8.4 cm and 1.3 original container, with the same base dimensions. cm and the height of the base is Use 1.25S to find h. . The height of the trapezoidal prism is 45 cm.

The total volume of the tray is the sum of the volumes of the two prisms. inches

ANSWER: 15 in; 15.4 in.

ANSWER: 14,735 cm³

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

26. 27.

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 a right . The total volume of the solid and the height is 11 cm. The base of the other prism is the sum of the volumes of the two rectangular is 4 cm by 3 cm and the height is 5 cm. prisms.

ANSWER: ANSWER: 225 cm3 120 m3

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29. 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?

28.

SOLUTION: The solid is a combination of a rectangular prism and SOLUTION: a half cylinder. The volume of the smaller can is

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

The volume of this combination shape is 713.1 yd³.

ANSWER: 713.1 in3

The height of the new can will be 35.1 cm.

ANSWER: 35.1 cm

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Find each measure to the nearest tenth. 31. A cylinder has a surface area of 144π square inches 30. 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 SOLUTION: centimeters. What is the height? Use the surface area formula to solve for r. SOLUTION:

ANSWER: 5.7 cm The radius is 6. Find the volume.

ANSWER: 678.6 in3

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32. A rectangular prism has a surface area of 432 square Find the volume of the solid formed by each net. 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. 33.

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. Find the volume. 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: 576 in3 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: 3934.9 cm3

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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? 34. SOLUTION: 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. a. First calculate the volume of soil in the pot. Then divide The radius is 1.8 m, the height is 4.8 m, and the slant the weight of the soil by the volume. 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 area are same. 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: 48.9 m3 b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plant should grow fairly well. 35. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will c. grow in it. The density of the soil sample is the ratio of its weight to its volume.

ANSWER: 3 a. If the weight of the container with the soil is 20 pounds a. 0.0019 lb / in eSolutions Manual - Powered by Cognero Page 13 11-2 Volumes of Prisms and Cylinders

b. The plant should grow well in this soil since the bulk density of 0.0019 lb / in3 is close to the desired bulk density of 0.0018 lb / in3. c. 8.3 lb Choose numbers for any two of the dimensions and we can solve for the third. Let = 2.25 in. and w = 36. 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:

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

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.

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: eSolutions Manual - Powered by Cognero Page 14 11-2 Volumes of Prisms and Cylinders

37. MODELING 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 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: 3,190,680.0 cm3

38. MULTISTEP Ryann is planning to build a sand castle. She wants her castle to be 4 feet high, 4 feet wide, and 6 feet deep. She has asked her brother Jack to bring sand over to her building site. They each have a bucket that is 8 inches in diameter and 16 inches tall. Each trip takes Jack about 30 seconds. a. After how long will Ryann have all of the sand she could possibly need to complete her castle? b. Describe your solution process. c. What assumptions did you make?

SOLUTION: a-b. The volume of the bucket is or about 804 in³. If Jack carries two buckets at a time, then he can carry 2(804) or 1608 in³. After converting the dimensions of the castle to inches, she finds the maximum volume of the castle is . Therefore, it would take Jack or 104 trips, to round up for eSolutions Manual - Powered by Cognero Page 15 11-2 Volumes of Prisms and Cylinders

the extra sand. 39. Find the volume of the regular pentagonal prism by If each trip takes him 30 seconds (or half a minute), dividing it into five equal triangular prisms. Describe this would take him , or 52 minutes, to the base area and height of each triangular prism. provide her with the maximum amount of sand that she needs.

c. The bucket is a cylinder. Ryann’s building site had no sand to start with. Jack goes at the same pace for every trip and doesn’t take any breaks. Each bucket is filled to the top and not overflowing with sand (or the average bucket-full equals the full capacity of the SOLUTION: bucket). Jack uses both buckets at the same time. The base of the prism can be divided into 5 congruent Ryann never stops to help Jack. Ryann needs enough triangles of a base 8 cm and the corresponding height sand to fill the entire volume. 5.5 cm. So, the pentagonal prism is a combination of ANSWER: 5 triangular prisms of height 10 cm. Find the base a. 52 min. area of each triangular prism. b. The volume of the bucket is 256π or about 804 in³. If Jack carries two buckets at a time, then he can carry 1608 in³. The maximum volume of the castle is Therefore, the volume of the pentagonal prism is 165,888 in³. Therefore, it would take Jack 104 trips, or 52 minutes, to provide her with the maximum amount of sand that she needs. ANSWER: c. The bucket is a cylinder. Ryann’s building site had 1100 cm3; Each triangular prism has a base area of no sand to start with. Jack goes at the same pace for every trip and doesn’t take any breaks. Each bucket or 22 cm2 and a height of 10 cm. is filled to the top and not overflowing with sand (or the average bucket-full equals the full capacity of the bucket). Jack uses both buckets at the same time. Ryann never stops to help Jack. Ryann needs enough sand to fill the entire volume.

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40. PATIOS Mr. Thomas is planning to remove an old SOLUTION: patio and install a new rectangular concrete patio 20 a. The oblique cylinder should look like the right feet long, 12 feet wide, and 4 inches thick. One cylinder (same height and size), except that it is contractor bid $2225 for the project. A second pushed a little to the side, like a slinky. 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 b. Find the volume of each. 4 in. = yd

Find the volume.

The volume of the square prism is greater. The total cost for the second contractor is about . 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 the b. VERBAL A square prism has a height of 10 volume x2 times greater. 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. c. ANALYTICAL Describe which change affects ANSWER: the volume of the cylinder more: multiplying the a. height by x or multiplying the radius by x. Explain. eSolutions Manual - Powered by Cognero Page 17 11-2 Volumes of Prisms and Cylinders

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. triangular prism with a right triangle as the base b. Greater than; a square with a side length of 6 m has an area of 36 m2. 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. SOLUTION: c. Multiplying the radius by x ; since the volume is The volume of the can is 20π in3. It takes three full represented by π r 2 h, multiplying the height by x cans to fill the container, so the volume of the makes the volume x times greater. Multiplying the container is 60π in3. radius by x makes the volume x 2 times greater,

assuming x > 1. a. Choose some basic values for 2 of the sides, and 42. ERROR ANALYSIS Franciso and Valerie each then determine the third side. Base: 3 by 5. 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. 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: Francisco is correct. 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- 5 by 5 by half the apothem multiplied by the perimeter of the base. c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. 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 The cylindrical can shown is used to eSolutions Manual - Powered by Cognero Page 18 11-2 Volumes of Prisms and Cylinders

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

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. 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: 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.

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.

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46. CONSTRUCT ARGUMENTS Determine whether 48. The rectangular prism shown here has a square base the following statement is true or false. Explain. and a volume of 132.3 cubic inches. Two cylinders with the same height and the same lateral area must have the same volume.

SOLUTION: The statement "Two cylinders with the same height and the same lateral area must have the same What is the perimeter of the base? volume. A 30 in. " is true. If two cylinders have the same height (h = 1 B 17.64 in. h2) and the same lateral area (L1 = L2), the circular C 16.8 in. bases must have the same area. D 4.2 in.

SOLUTION:

The radii must also be equal. Substitute the given information into the formula for a ANSWER: rectangular prism and solve for x, one side of the True; if two cylinders have the same height and the square base. same lateral area, the circular bases must have the same area. Therefore, πr2h is the same for each cylinder.

47. WRITING IN MATH How are the volume formulas for prisms and cylinders similar? How are they different? If the length of one side of the square base is 4.2 SOLUTION: inches, then the perimeter is 4(4.2) or 16.8 inches. Both formulas involve multiplying the area of the base The correct choice is C. by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on ANSWER: the type of polygon it is. The base of a cylinder is a C 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 base of a cylinder is a circle, so its area is πr2.

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49. An aquarium is a rectangular prism that is 20 inches 50. Scott adds sand to the cylindrical container shown long, 1 foot wide, and 15 inches tall. Denise fills the below so that the surface of the sand is 2 inches aquarium using a container that holds 400 cubic below the top of the container. inches of water. Assuming she always fills the container completely, how many times will Denise need to pour water from the container into the aquarium?

SOLUTION: Begin by sketching a figure and label it with the given information. Which of the following is the best estimate of the volume of the sand in the container? A 127 in³ B 269 in³ C 318 in³ D 445 in³ E 1272 in³

SOLUTION: One way to find the volume of sand that fills the cylinder to a height 2 inches below the the top of the container, subtract 2 inches from the height and find the volume of this shorter cylinder. Find the volume of the prism:

The correct choice is C. She will need containers of 400 in³ water to fill up the aquarium. ANSWER: C ANSWER: 9

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51. A cylindrical tank used for oil storage has a height 52. The cylindrical can of juice shown here has a volume that is half the length of its radius. If the volume of of 300 cubic centimeters. What is the diameter of the the tank is 1,122,360 ft³, what is the tank’s radius in can in centimeters? Round to the nearest tenth. feet? Round to the nearest tenth.

SOLUTION: Sketch a cylinder where the height is half the length of the radius.

A 6.2 cm B 3.1 cm C 9.8 cm D 30 cm Use these dimensions and the given volume to solve for the length of the radius of the tank. SOLUTION: Use the formula for the volume of a cylinder to find the radius, then double the radius to find the height.

ANSWER: 89.4 So, the diameter is about 6.18 centimeters. The correct choice is A.

ANSWER: A

53. A red has an edge length of 2 inches. A blue cube has an edge length that is double that of the red cube. What is the volume of the blue cube?

SOLUTION: A blue cube has an edge length that is double that of the red cube or 2(2 inches) = 4 inches.

The volume of the blue cube is (4 inches)3 = 64 in.3

ANSWER: 64 in.3

54. MULTI-STEP Kara has a cylindrical pillar candle that is 4 inches in diameter and 9 inches tall. She melts the candle and pours all of the wax into a eSolutions Manual - Powered by Cognero Page 22 11-2 Volumes of Prisms and Cylinders

square mold that is 4 inches on each side. 7.0625 = h

The square candle is about 7 inches tall.

d. To make the square candle the same height as the cylindrical candle (9 inches), Kara would need 9(4)2 – 113 = 31 in.3of wax.

ANSWER: a. 113 in.3 a. To the nearest cubic inch, what is the volume of the pillar candle? b. 113 = (4)2h.

b. Write an equation that makes the volume of the c. 7 in. pillar candle equal to the volume of a square candle with 4-inch sides and an unknown height. d. 31 in.3 c. Solve the equation to find the height of the square candle.

d. If Kara wanted to make the square candle the same height as the cylindrical candle (9 inches), how much more wax would she need?

SOLUTION: MULTI-STEP Kara has a cylindrical pillar candle that is 4 inches in diameter and 9 inches tall. She melts the candle and pours all of the wax into a square mold that is 4 inches on each side.

a. The volume of the pillar candle can be found using the formula for the volume of a cylinder.

V = πr2h V = π(2 in.)2(9 in.) V = 36π in.3 V ≈ 113 in.3

b. An equation with the volume of the pillar candle equal to the volume of a square candle with 4-inch sides and an unknown height is 113 = (4)2h.

c. Solve the equation to find the height of the square candle.

113 = (4)2h eSolutions Manual - Powered by Cognero Page 23