Find the Volume of Each Pyramid. 1. SOLUTION: the Volume of A

1. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

2. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

12-5 Volumes of Pyramids and Cones

Find the volume of each pyramid. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

2. SOLUTION:

The volume of a pyramid is , where B is the Find the volume of each cone. Round to the area of the base and h is the height of the pyramid. nearest tenth. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters. 5.

SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches. 3. a rectangular pyramid with a height of 5.2 meters eSolutionsand Manuala base -8Powered metersby byCognero 4.5 meters Page 1

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

6. SOLUTION:

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: Use trigonometry to find the radius r. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

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

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters 5. SOLUTION: SOLUTION: The volume of a circular cone is , or The volume of a circular cone is , or , where B is the area of the base, h is the , where B is the area of the base, h is the height of the cone, and r is the radius of the base. height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the

Since the diameter of this cone is 7 inches, the radius height is 10.5 millimeters.

is or 3.5 inches. The height of the cone is 4 inches.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

Use trigonometry to find the radius r.

The volume of a circular cone is , or So, the height of the cone is 20 meters.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of 7. an oblique cone with a height of 10.5 millimeters and air that the heating and cooling systems would have a radius of 1.6 millimeters to accommodate. Round to the nearest tenth. SOLUTION: SOLUTION:

The volume of a circular cone is , or The volume of a circular cone is , where B is the area of the base and h is the height of the , where B is the area of the base, h is the cone. height of the cone, and r is the radius of the base. For this cone, the area of the base is 15,400 square The radius of this cone is 1.6 millimeters and the feet and the height is 100 feet. height is 10.5 millimeters.

CCSS SENSE-MAKING Find the volume of 8. a cone with a slant height of 25 meters and a radius each pyramid. Round to the nearest tenth if of 15 meters necessary. SOLUTION:

10. SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

So, the height of the cone is 20 meters.

11.

9. MUSEUMS The sky dome of the National Corvette SOLUTION: Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the The volume of a pyramid is , where B is the base is about 15,400 square feet, find the volume of area of the base and h is the height of the pyramid. air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. CCSS SENSE-MAKING Find the volume of

each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. 13.

SOLUTION: The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square area of the base and h is the height of the pyramid. feet and an altitude of 7 feet

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and SOLUTION: hypotenuse 10.2 centimeters The volume of a pyramid is , where B is the SOLUTION: Find the height of the right triangle. base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION: 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has The volume of a pyramid is , where B is the a volume of 144 cubic centimeters. Find the height. area of the base and h is the height of the pyramid. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the height of the triangular pyramid is 18 cm.

16. A triangular pyramid with a right triangle base with a Find the volume of each cone. Round to the leg 8 centimeters and hypotenuse 10 centimeters has nearest tenth. a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. 17. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the SOLUTION: triangle. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 3 cm. Therefore, the volume of the cone is about 235.6 in .

18. SOLUTION: So, the area of the base B is 24 cm2. The volume of a circular cone is , where Replace V with 144 and B with 24 in the formula for r is the radius of the base and h is the height of the the volume of a pyramid and solve for the height h. cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 Therefore, the volume of the cone is about 134.8 3 cm. cm .

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

19. 17. SOLUTION: SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 Use a trigonometric ratio to find the height h of the cone. inches.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the

3 cone. The radius of this cone is 8 centimeters. Therefore, the volume of the cone is about 235.6 in .

18.

SOLUTION: Therefore, the volume of the cone is about 1473.1 cm3. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

20. SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm . Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

The volume of a circular cone is , where 21. an oblique cone with a diameter of 16 inches and an r is the radius of the base and h is the height of the altitude of 16 inches cone. The radius of this cone is 8 centimeters. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1473.1 cm3.

Therefore, the volume of the cone is about 1072.3 20. in 3.

SOLUTION: 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft . 3 Therefore, the volume of the cone is about 5.8 cm . 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a SOLUTION: paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest The volume of a circular cone is , where tenth. r is the radius of the base and h is the height of the SOLUTION: cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter Therefore, the paper cone will hold about 234.6 cm3 SOLUTION: of roasted peanuts. The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean 24. CCSS MODELING The Pyramid Arena in Theorem to find the height h of the cone. Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

3 Therefore, the volume of the cone is about 5.8 cm .

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest SOLUTION: tenth. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a circular cone is , where The base of the pyramid is a regular octagon with r is the radius of the base and h is the height of the sides of 2 feet. A central angle of the octagon is cone. Since the diameter of the cone is 8 or 45°, so the angle formed in the triangle below is 22.5°. centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Use a trigonometric ratio to find the apothem a. Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

Find the volume of each solid. Round to the 25. GARDENING The greenhouse is a regular nearest tenth. octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Use a trigonometric ratio to find the apothem a. 27.

SOLUTION:

The height of this pyramid is 5 feet.

28. Therefore, the volume of the greenhouse is about SOLUTION: 3 32.2 ft .

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

26. SOLUTION: 29. HEATING Sam is building an art studio in her Volume of the solid given = Volume of the small backyard. To buy a heating unit for the space, she cone + Volume of the large cone needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per

cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

27.

SOLUTION: The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the 28. volume of the model. Explain why knowing the SOLUTION: volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) It tells Marta how much clay is needed to make the required to heat the building. For new construction model. with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to 31. CHANGING DIMENSIONS A cone has a radius purchase? of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: SOLUTION: Find the volume of the original cone. Then alter the The building can be broken down into the rectangular values. base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is a. Double h.

The total volume is therefore 5000 + 1666.67 = 3 The volume is doubled. 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is b. Double r. 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation.

SOLUTION: 2 The volume of a pyramid is , where B is the The volume is multiplied by 2 or 4.

area of the base and h is the height of the pyramid. c. Double r and h.

It tells Marta how much clay is needed to make the 3 model. volume is multiplied by 2 or 8.

31. CHANGING DIMENSIONS A cone has a radius Find each measure. Round to the nearest tenth of 4 centimeters and a height of 9 centimeters. if necessary. Describe how each change affects the volume of the 32. A square pyramid has a volume of 862.5 cubic cone. centimeters and a height of 11.5 centimeters. Find the side length of the base. a. The height is doubled. b. The radius is doubled. SOLUTION: c. Both the radius and the height are doubled. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. Find the volume of the original cone. Then alter the Let s be the side length of the base. values.

a. Double h.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the The volume is doubled. height is 12 inches. What is the diameter?

b. Double r. SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters. 2 The volume is multiplied by 2 or 4.

c. Double r and h.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth The diameter is 2(7) or 14 inches. if necessary. 32. A square pyramid has a volume of 862.5 cubic 34. The lateral area of a cone is 71.6 square millimeters centimeters and a height of 11.5 centimeters. Find and the slant height is 6 millimeters. What is the the side length of the base. volume of the cone? SOLUTION: SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the So, the radius is about 3.8 millimeters. height is 12 inches. What is the diameter? Use the Pythagorean Theorem to find the height of SOLUTION: the cone.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters Therefore, the volume of the cone is about 70.2 and the slant height is 6 millimeters. What is the 3 volume of the cone? mm . SOLUTION: 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. The lateral area of a cone is , where r is SOLUTION: the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the a. Use rectangular bases and pick values that radius r. multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

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

So, the height of the cone is about 4.64 millimeters. b. The volumes are the same. The volume of a pyramid equals one third times the base area times The volume of a circular cone is , where the height. So, if the base areas of two pyramids are r is the radius of the base and h is the height of the equal and their heights are equal, then their volumes cone. are equal.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with c. If the base area is multiplied by 5, the volume is different bases that have a height of 10 centimeters multiplied by 5. If the height is multiplied by 5, the and a base area of 24 square centimeters. volume is multiplied by 5. If both the base area and b. VERBAL What is true about the volumes of the the height are multiplied by 5, the volume is multiplied two pyramids that you drew? Explain. by 5 · 5 or 25. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes 36. CCSS ARGUMENTS Determine whether the are equal. following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

SOLUTION: Alexandra incorrectly used the slant height. The volume of a cone with a radius r and height h is . The volume of a prism with a height of 38. REASONING A cone has a volume of 568 cubic h is where B is the area of the base of the centimeters. What is the volume of a cylinder that prism. Set the volumes equal. has the same radius and height as the cone? Explain your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning.

SOLUTION: The volumes will only be equal when the radius of The formula for volume of a prism is V = Bh and the the cone is equal to or when . formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, Therefore, the statement is true sometimes if the then in order to have the same volume, the height of base area of the cone is 3 times as great as the base the pyramid must be 3 times as great as the height of area of the prism. For example, if the base of the the prism. prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of Set the base areas of the prism and pyramid, and them correct? Explain your answer. make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: SOLUTION: The slant height is used for surface area, but the To find the volume of each solid, you must know the height is used for volume. For this cone, the slant area of the base and the height. The volume of a height of 13 is provided, and we need to calculate the pyramid is one third the volume of a prism that has height before we can calculate the volume. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area. Alexandra incorrectly used the slant height. 41. A conical sand toy has the dimensions as shown 38. REASONING A cone has a volume of 568 cubic below. How many cubic centimeters of sand will it centimeters. What is the volume of a cylinder that hold when it is filled to the top? has the same radius and height as the cone? Explain your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone A 12π B 15π is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same C radius and height.

D 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same SOLUTION: volume. Explain your reasoning. Use the Pythagorean Theorem to find the radius r of SOLUTION: the cone. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the

height of the prism. So, the radius of the cone is 3 centimeters.

Sample answer: The volume of a circular cone is , where A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 r is the radius of the base and h is the height of the and a height of 4. cone.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has Therefore, the correct choice is A. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the 42. SHORT RESPONSE Brooke is buying a tent that same height and base area. is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, 41. A conical sand toy has the dimensions as shown how tall is the tent’s center pole? below. How many cubic centimeters of sand will it hold when it is filled to the top? SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

A 12π B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of

the cone. 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters. J The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. Possible outcomes: {6 red, 4 blue, 5 orange, 10 green}

Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, So, the correct choice is F. how tall is the tent’s center pole? SOLUTION: 44. SAT/ACT For all

The volume of a pyramid is , where B is the A –8 B x – 4 area of the base and h is the height of the pyramid. C D E

SOLUTION:

43. PROBABILITY A spinner has sections colored So, the correct choice is E. red, blue, orange, and green. The table below shows the results of several spins. What is the experimental Find the volume of each prism. probability of the spinner landing on orange?

45. SOLUTION:

F The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 G inches and a width of 12 inches. The height h of the prism is 6 inches. H

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} 3 Number of possible outcomes : 25 Therefore, the volume of the prism is 1008 in . Favorable outcomes: {5 orange} Number of favorable outcomes: 5

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The So, the correct choice is F. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will 44. SAT/ACT For all bisect the base. Use the Pythagorean Theorem to find the height of the triangle. A –8 B x – 4 C D E

SOLUTION:

So, the correct choice is E. So, the height of the triangle is 12 feet. Find the area of the triangle. Find the volume of each prism.

45. SOLUTION: 2 The volume of a prism is , where B is the So, the area of the base B is 60 ft . area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 The height h of the prism is 19 feet. inches and a width of 12 inches. The height h of the prism is 6 inches.

3 Therefore, the volume of the prism is 1140 ft .

3 Therefore, the volume of the prism is 1008 in .

47. SOLUTION: 46. The volume of a prism is , where B is the area of the base and h is the height of the prism. The SOLUTION: base of this prism is a rectangle with a length of 79.4 The volume of a prism is , where B is the meters and a width of 52.5 meters. The height of the area of the base and h is the height of the prism. The prism is 102.3 meters. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. So, the height of the triangle is 12 feet. Find the area of the triangle. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

So, the area of the base B is 60 ft2. Find the slant height of the conical top.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1140 ft .

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 47.

SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

The formula for finding the surface area of a cylinder Therefore, the volume of the prism is about 426,437.6 is , where is r is the radius and h is the slant 3 height of the cylinder. m .

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin Surface area of the bin = Surface area of the allows the grain to be emptied more easily. Use the cylinder + Surface area of the conical top + surface dimensions in the diagram to find the entire surface area of the conical bottom. area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find Find the area of each shaded region. Polygons the surface area of the cylinder and add them. in 50 - 52 are regular. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. 49. Find the slant height of the conical top. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2)

or 9 ft. 50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the The formula for finding the surface area of a cylinder angle, so we use 30° when using trig to find the is , where is r is the radius and h is the slant length of the apothem. height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50 Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the Find the area of the circle with a radius of 3.6 feet. angle, so we use 30° when using trig to find the length of the apothem.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three

isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Use a trigonometric ratio to find the value of b. Now find the area of the hexagon.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet. 51.

SOLUTION: Subtract to find the area of shaded region.

The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 2 26.6 ft .

So, the area of the circle is about 40.7 square feet.

circle. The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown Use a trigonometric ratio to find the value of b. below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle. Use trigonometric ratios to find the values of b and

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A Therefore, the area of the shaded region is about (equilateral triangle) + A(inscribed circle) 2 26.6 ft .

52. Therefore, the area of the shaded region is about SOLUTION: 168.2 mm2. The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid.

1. 2. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs The base of this pyramid is a regular pentagon with of 9 inches and 5 inches and the height of the sides of 4.4 centimeters and an apothem of 3 pyramid is 10 inches. centimeters. The height of the pyramid is 12 centimeters.

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the 2. area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of The volume of a pyramid is , where B is the the pyramid is 5.2 meters. area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 3. a rectangular pyramid with a height of 5.2 meters meters. The height of the pyramid is 14 meters. and a base 8 meters by 4.5 meters SOLUTION:

5. SOLUTION: 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths The volume of a circular cone is , or SOLUTION: , where B is the area of the base, h is the The volume of a pyramid is , where B is the height of the cone, and r is the radius of the base. area of the base and h is the height of the pyramid. Since the diameter of this cone is 7 inches, the radius The base of this pyramid is a square with sides of 8 is or 3.5 inches. The height of the cone is 4 inches.

meters. The height of the pyramid is 14 meters.

12-5 Volumes of Pyramids and Cones

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

6. SOLUTION: 5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the Use trigonometry to find the radius r. height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

6. 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the

Use trigonometry to find the radius r. height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: eSolutions Manual - Powered by Cognero Page 2

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: Use the Pythagorean Theorem to find the height h of The volume of a circular cone is , or the cone. Then find its volume.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

So, the height of the cone is 20 meters.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth.

Use the Pythagorean Theorem to find the height h of SOLUTION: the cone. Then find its volume. The volume of a circular cone is , where B

is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

So, the height of the cone is 20 meters.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. 10. SOLUTION: SOLUTION:

The volume of a circular cone is , where B The volume of a pyramid is , where B is the is the area of the base and h is the height of the area of the base and h is the height of the pyramid. cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. 11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 10.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

12. The base is a hexagon, so we need to make a right tri SOLUTION: determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl The volume of a pyramid is , where B is the 90° triangle.

area of the base and h is the height of the pyramid.

The apothem is . 13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle. 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is . 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and

hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters The volume of a pyramid is , where B is the SOLUTION: Find the height of the right triangle. area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The length of the other leg of the right triangle is 6 16. A triangular pyramid with a right triangle base with a cm. leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle. So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

Therefore, the height of the triangular pyramid is 18 cm. The length of the other leg of the right triangle is 6 cm. Find the volume of each cone. Round to the nearest tenth.

17. SOLUTION: So, the area of the base B is 24 cm2. The volume of a circular cone is ,

Replace V with 144 and B with 24 in the formula for where r is the radius of the base and h is the height the volume of a pyramid and solve for the height h. of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Therefore, the height of the triangular pyramid is 18 cm.

Find the volume of each cone. Round to the Therefore, the volume of the cone is about 235.6 in3. nearest tenth.

17. 18. SOLUTION: SOLUTION: The volume of a circular cone is , The volume of a circular cone is , where where r is the radius of the base and h is the height of the cone. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. Since the diameter of this cone is 10 inches, the

radius is or 5 inches. The height of the cone is 9 inches.

Therefore, the volume of the cone is about 134.8 3 cm .

Therefore, the volume of the cone is about 235.6 in3.

19.

18. SOLUTION: SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Use a trigonometric ratio to find the height h of the cone.

Therefore, the volume of the cone is about 134.8 The volume of a circular cone is , where 3 cm . r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

19. SOLUTION:

Therefore, the volume of the cone is about 1473.1 cm3.

Use a trigonometric ratio to find the height h of the 20. cone. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1473.1 The volume of a circular cone is , where 3 cm . r is the radius of the base and h is the height of the cone.

20. SOLUTION:

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches Use trigonometric ratios to find the height h and the radius r of the cone. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: 3 Therefore, the volume of the cone is about 2.8 ft . The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean 21. an oblique cone with a diameter of 16 inches and an Theorem to find the height h of the cone. altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter

SOLUTION: The cone has a radius r of 1 centimeter and a slant 3 Therefore, the volume of the cone is about 5.8 cm . height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts. 3 Therefore, the volume of the cone is about 5.8 cm . 24. CCSS MODELING The Pyramid Arena in 23. SNACKS Approximately how many cubic Memphis, Tennessee, is the third largest pyramid in centimeters of roasted peanuts will completely fill a the world. It is approximately 350 feet tall, and its paper cone that is 14 centimeters high and has a base square base is 600 feet wide. Find the volume of this diameter of 8 centimeters? Round to the nearest pyramid. tenth. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a circular cone is , where area of the base and h is the height of the pyramid. r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this

pyramid. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse? Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet. SOLUTION:

Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

Use a trigonometric ratio to find the apothem a.

26. SOLUTION:

Volume of the solid given = Volume of the small

The height of this pyramid is 5 feet. cone + Volume of the large cone

Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

27. SOLUTION: 26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

28. SOLUTION:

27.

SOLUTION:

28. SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase? The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the SOLUTION: volume is helpful in this situation. The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base SOLUTION: is The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume of the ceiling is

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius

of 4 centimeters and a height of 9 centimeters. The total volume is therefore 5000 + 1666.67 = 3 Describe how each change affects the volume of the 6666.67 ft . Two BTU's are needed for every cubic cone. foot, so the size of the heating unit Sam should buy is a. The height is doubled. 6666.67 × 2 = 13,333 BTUs. b. The radius is doubled.

c. Both the radius and the height are doubled. 30. SCIENCE Refer to page 825. Determine the SOLUTION: volume of the model. Explain why knowing the Find the volume of the original cone. Then alter the

volume is helpful in this situation. values. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

a. Double h.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the The volume is doubled. cone. a. The height is doubled. b. Double r. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

a. Double h.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary.

The volume is doubled. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. b. Double r. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

3 volume is multiplied by 2 or 8. The side length of the base is 15 cm. Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic 33. The volume of a cone is 196π cubic inches and the centimeters and a height of 11.5 centimeters. Find height is 12 inches. What is the diameter? the side length of the base. SOLUTION: SOLUTION: The volume of a circular cone is , or The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. , where B is the area of the base, h is the Let s be the side length of the base. height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm. The diameter is 2(7) or 14 inches.

33. The volume of a cone is 196π cubic inches and the 34. The lateral area of a cone is 71.6 square millimeters height is 12 inches. What is the diameter? and the slant height is 6 millimeters. What is the SOLUTION: volume of the cone? SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The diameter is 2(7) or 14 inches. So, the radius is about 3.8 millimeters. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the Use the Pythagorean Theorem to find the height of

volume of the cone? the cone. SOLUTION:

So, the height of the cone is about 4.64 millimeters.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. The volume of a circular cone is , where Replace L with 71.6 and with 6, then solve for the r is the radius of the base and h is the height of the radius r. cone.

So, the radius is about 3.8 millimeters. Therefore, the volume of the cone is about 70.2 3 Use the Pythagorean Theorem to find the height of mm . the cone. 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base So, the height of the cone is about 4.64 millimeters. area and/or the height of the pyramid by 5 affects the volume of the pyramid. The volume of a circular cone is , where SOLUTION: a. Use rectangular bases and pick values that r is the radius of the base and h is the height of the multiply to make 24. cone.

Sample answer:

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. b. The volumes are the same. The volume of a SOLUTION: pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are a. Use rectangular bases and pick values that equal and their heights are equal, then their volumes multiply to make 24. are equal.

Sample answer:

The volumes will only be equal when the radius of 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never the cone is equal to or when . true. Justify your reasoning. Therefore, the statement is true sometimes if the The volume of a cone with radius r and height h base area of the cone is 3 times as great as the base equals the volume of a prism with height h. area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume SOLUTION: is 10h cubic units. So, the cone must have a base The volume of a cone with a radius r and height h is area of 30 square units so that its volume is . The volume of a prism with a height of or 10h cubic units. h is where B is the area of the base of the prism. Set the volumes equal. 37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

The volumes will only be equal when the radius of SOLUTION: the cone is equal to or when . The slant height is used for surface area, but the height is used for volume. For this cone, the slant Therefore, the statement is true sometimes if the height of 13 is provided, and we need to calculate the base area of the cone is 3 times as great as the base height before we can calculate the volume. area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base Alexandra incorrectly used the slant height. area of 30 square units so that its volume is 38. REASONING A cone has a volume of 568 cubic or 10h cubic units. centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain 37. ERROR ANALYSIS Alexandra and Cornelio are your reasoning. calculating the volume of the cone below. Is either of SOLUTION: them correct? Explain your answer. 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same

radius and height. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning.

SOLUTION: SOLUTION: The formula for volume of a prism is V = Bh and the The slant height is used for surface area, but the formula for the volume of a pyramid is one-third of height is used for volume. For this cone, the slant that. So, if a pyramid and prism have the same base, height of 13 is provided, and we need to calculate the then in order to have the same volume, the height of height before we can calculate the volume. the pyramid must be 3 times as great as the height of the prism.

Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning.

SOLUTION: Set the base areas of the prism and pyramid, and 3 make the height of the pyramid equal to 3 times the 1704 cm ; The formula for the volume of a cylinder height of the prism. is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times Sample answer: as much as the volume of a cone with the same A square pyramid with a base area of 16 and a radius and height. height of 12, a prism with a square base area of 16 and a height of 4. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same 40. WRITING IN MATH Compare and contrast volume. Explain your reasoning. finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: The formula for volume of a prism is V = Bh and the SOLUTION: formula for the volume of a pyramid is one-third of To find the volume of each solid, you must know the that. So, if a pyramid and prism have the same base, area of the base and the height. The volume of a then in order to have the same volume, the height of pyramid is one third the volume of a prism that has the pyramid must be 3 times as great as the height of the same height and base area. The volume of a the prism. cone is one third the volume of a cylinder that has the same height and base area. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. A 12π

B 15π Sample answer:

A square pyramid with a base area of 16 and a C height of 12, a prism with a square base area of 16

and a height of 4. D 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding SOLUTION: volumes of prisms and cylinders. Use the Pythagorean Theorem to find the radius r of the cone. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π So, the radius of the cone is 3 centimeters. B 15π The volume of a circular cone is , where C r is the radius of the base and h is the height of the cone. D

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole?

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange? Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole?

SOLUTION: F

The volume of a pyramid is , where B is the G

area of the base and h is the height of the pyramid. H

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

43. PROBABILITY A spinner has sections colored

red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the correct choice is F.

F 44. SAT/ACT For all

G A –8 B x – 4

H C D J E SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 SOLUTION: green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

So, the correct choice is E. Find the volume of each prism.

45. So, the correct choice is F. SOLUTION: 44. SAT/ACT For all The volume of a prism is , where B is the area of the base and h is the height of the prism. The A –8 base of this prism is a rectangle with a length of 14 B x – 4 inches and a width of 12 inches. The height h of the C prism is 6 inches.

D E

SOLUTION:

3 Therefore, the volume of the prism is 1008 in .

So, the correct choice is E.

Find the volume of each prism. 46. SOLUTION: The volume of a prism is , where B is the 45. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base SOLUTION: of 10 feet and two legs of 13 feet. The height h will The volume of a prism is , where B is the bisect the base. Use the Pythagorean Theorem to area of the base and h is the height of the prism. The find the height of the triangle. base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

3 Therefore, the volume of the prism is 1008 in .

So, the height of the triangle is 12 feet. Find the area of the triangle.

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base

of 10 feet and two legs of 13 feet. The height h will 2 bisect the base. Use the Pythagorean Theorem to So, the area of the base B is 60 ft . find the height of the triangle. The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1140 ft .

So, the height of the triangle is 12 feet. Find the area 47. of the triangle. SOLUTION:

The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination

hopper cone and bin used by farmers to store grain 3 Therefore, the volume of the prism is 1140 ft . after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. 47. SOLUTION: SOLUTION: To find the entire surface area of the bin, find the The volume of a prism is , where B is the surface area of the conical top and bottom and find area of the base and h is the height of the prism. The the surface area of the cylinder and add them. base of this prism is a rectangle with a length of 79.4 The formula for finding the surface area of a cone is meters and a width of 52.5 meters. The height of the , where is r is the radius and l is the slant height prism is 102.3 meters. of the cone.

Find the slant height of the conical top.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the Find the slant height of the conical bottom. nearest square foot. The height of the conical bottom is 28 – (5 + 12 + 2) Refer to the photo on Page 847. or 9 ft.

SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the slant height of the conical bottom.

The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle The formula for finding the surface area of a cylinder Area of the rectangle = 10(5) is , where is r is the radius and h is the slant = 50 height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

50. SOLUTION: Area of the shaded region = Area of the circle – Find the area of each shaded region. Polygons Area of the hexagon in 50 - 52 are regular.

49. SOLUTION: A regular hexagon has 6 sides, so the measure of the Area of the shaded region = Area of the rectangle – Area of the circle interior angle is . The apothem bisects the Area of the rectangle = 10(5) angle, so we use 30° when using trig to find the = 50 length of the apothem.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem. Now find the area of the hexagon.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

Now find the area of the hexagon.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the 51. height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle. SOLUTION:

The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

Use a trigonometric ratio to find the value of b.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

The equilateral triangle can be divided into three isosceles triangles with central angles of or So, the area of the equilateral triangle is about 67.3 120°, so the angle in the triangle created by the square feet. height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle. Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 Use a trigonometric ratio to find the value of b. 26.6 ft .

52. SOLUTION: Now use the area of the isosceles triangles to find The area of the shaded region is the area of the the area of the equilateral triangle. circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the Therefore, the area of the shaded region is about isosceles triangle is 60° and the base is half the 2 length of the side of the equilateral triangle as shown 26.6 ft . below.

52.

SOLUTION: The area of the shaded region is the area of the Use trigonometric ratios to find the values of b and circumscribed circle minus the area of the equilateral h. triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

The area of the equilateral triangle will equal three Divide the equilateral triangle into three isosceles times the area of any of the isosceles triangles. Find triangles with central angles of or 120°. The the area of the shaded region. angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the A(shaded region) = A(circumscribed circle) - A length of the side of the equilateral triangle as shown (equilateral triangle) + A(inscribed circle) below.

Therefore, the area of the shaded region is about 2 168.2 mm . Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

Find the volume of each pyramid. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

1. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length The volume of a pyramid is , where B is the of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters. area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

2. SOLUTION:

5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius 3. a rectangular pyramid with a height of 5.2 meters is or 3.5 inches. The height of the cone is 4 inches. and a base 8 meters by 4.5 meters SOLUTION:

6. SOLUTION:

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the Use trigonometry to find the radius r.

area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

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

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

6. SOLUTION:

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

Use trigonometry to find the radius r.

The volume of a circular cone is , or So, the height of the cone is 20 meters. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the 7. an oblique cone with a height of 10.5 millimeters and base is about 15,400 square feet, find the volume of a radius of 1.6 millimeters air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION: SOLUTION: The volume of a circular cone is , or The volume of a circular cone is , where B , where B is the area of the base, h is the is the area of the base and h is the height of the height of the cone, and r is the radius of the base. cone. The radius of this cone is 1.6 millimeters and the For this cone, the area of the base is 15,400 square height is 10.5 millimeters. feet and the height is 100 feet.

12-5 Volumes of Pyramids and Cones

8. a cone with a slant height of 25 meters and a radius CCSS SENSE-MAKING Find the volume of of 15 meters each pyramid. Round to the nearest tenth if necessary. SOLUTION:

10.

Use the Pythagorean Theorem to find the height h of SOLUTION: the cone. Then find its volume. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

So, the height of the cone is 20 meters.

11. 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical SOLUTION: building. If the height is 100 feet and the area of the The volume of a pyramid is , where B is the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have area of the base and h is the height of the pyramid. to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

12. SOLUTION:

The volume of a pyramid is , where B is the CCSS SENSE-MAKING Find the volume of area of the base and h is the height of the pyramid. each pyramid. Round to the nearest tenth if eSolutionsnecessary.Manual - Powered by Cognero Page 3

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 13.

SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. 15. a triangular pyramid with a height of 4.8 centimeters SOLUTION: and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters The volume of a pyramid is , where B is the SOLUTION: base and h is the height of the pyramid. Find the height of the right triangle.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION: 16. A triangular pyramid with a right triangle base with a The volume of a pyramid is , where B is the leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. area of the base and h is the height of the pyramid. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the height of the triangular pyramid is 18 cm. 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has Find the volume of each cone. Round to the a volume of 144 cubic centimeters. Find the height. nearest tenth. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a 17. of the right triangle and then find the area of the triangle. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm. Therefore, the volume of the cone is about 235.6 in3.

18. So, the area of the base B is 24 cm2. SOLUTION:

Replace V with 144 and B with 24 in the formula for The volume of a circular cone is , where the volume of a pyramid and solve for the height h. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 Therefore, the volume of the cone is about 134.8 cm. 3 cm . Find the volume of each cone. Round to the nearest tenth.

19. 17. SOLUTION: SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 Use a trigonometric ratio to find the height h of the

inches. cone.

The volume of a circular cone is , where

3 r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 235.6 in . cone. The radius of this cone is 8 centimeters.

18. SOLUTION: Therefore, the volume of the cone is about 1473.1 The volume of a circular cone is , where cm3. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

20. SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm .

Use trigonometric ratios to find the height h and the radius r of the cone.

19.

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft . The volume of a circular cone is , where r is the radius of the base and h is the height of the 21. an oblique cone with a diameter of 16 inches and an

cone. The radius of this cone is 8 centimeters. altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1473.1 cm3.

20. Therefore, the volume of the cone is about 1072.3 in 3. SOLUTION: 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft . 3 21. an oblique cone with a diameter of 16 inches and an Therefore, the volume of the cone is about 5.8 cm . altitude of 16 inches 23. SNACKS Approximately how many cubic SOLUTION: centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base The volume of a circular cone is , where diameter of 8 centimeters? Round to the nearest tenth. r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, SOLUTION: the radius is or 8 inches. The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter

SOLUTION: Therefore, the paper cone will hold about 234.6 cm3 The cone has a radius r of 1 centimeter and a slant of roasted peanuts. height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

3 Therefore, the volume of the cone is about 5.8 cm .

SOLUTION: The volume of a pyramid is , where B is the The volume of a circular cone is , where area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with r is the radius of the base and h is the height of the sides of 2 feet. A central angle of the octagon is cone. Since the diameter of the cone is 8 or 45°, so the angle formed in the triangle centimeters, the radius is or 4 centimeters. The below is 22.5°. height of the cone is 14 centimeters.

Therefore, the paper cone will hold about 234.6 cm3 Use a trigonometric ratio to find the apothem a. of roasted peanuts.

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the volume of the greenhouse is about 3 32.2 ft . GARDENING 25. The greenhouse is a regular Find the volume of each solid. Round to the octagonal pyramid with a height of 5 feet. The base nearest tenth. has side lengths of 2 feet. What is the volume of the greenhouse?

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone SOLUTION:

Use a trigonometric ratio to find the apothem a. 27. SOLUTION:

The height of this pyramid is 5 feet.

Therefore, the volume of the greenhouse is about 28. 3 32.2 ft . SOLUTION:

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

required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

27. SOLUTION:

The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

28. 30. SCIENCE Refer to page 825. Determine the SOLUTION: volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction It tells Marta how much clay is needed to make the with good insulation, there should be 2 BTUs per model. cubic foot. What size unit does Sam need to purchase? 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled.

SOLUTION: SOLUTION: The building can be broken down into the rectangular Find the volume of the original cone. Then alter the

base and the pyramid ceiling. The volume of the base values. is

The volume of the ceiling is a. Double h.

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic The volume is doubled. foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. b. Double r.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the 2 The volume is multiplied by 2 or 4. area of the base and h is the height of the pyramid. c. Double r and h.

It tells Marta how much clay is needed to make the model. volume is multiplied by 23 or 8. 31. CHANGING DIMENSIONS A cone has a radius Find each measure. Round to the nearest tenth of 4 centimeters and a height of 9 centimeters. if necessary. Describe how each change affects the volume of the 32. A square pyramid has a volume of 862.5 cubic cone. centimeters and a height of 11.5 centimeters. Find a. The height is doubled. the side length of the base. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the Find the volume of the original cone. Then alter the area of the base and h is the height of the pyramid.

values. Let s be the side length of the base.

a. Double h.

The side length of the base is 15 cm.

The volume is doubled. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? b. Double r. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 2 The volume is multiplied by 2 or 4. centimeters.

c. Double r and h.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. The diameter is 2(7) or 14 inches. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find 34. The lateral area of a cone is 71.6 square millimeters the side length of the base. and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION: SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the So, the radius is about 3.8 millimeters. height is 12 inches. What is the diameter?

SOLUTION: Use the Pythagorean Theorem to find the height of the cone. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the Therefore, the volume of the cone is about 70.2 volume of the cone? 3 mm . SOLUTION: 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the The lateral area of a cone is , where r is volume of the pyramid. the radius and is the slant height of the cone. SOLUTION: Replace L with 71.6 and with 6, then solve for the a. radius r. Use rectangular bases and pick values that

multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

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

So, the height of the cone is about 4.64 millimeters.

b. The volumes are the same. The volume of a The volume of a circular cone is , where pyramid equals one third times the base area times r is the radius of the base and h is the height of the the height. So, if the base areas of two pyramids are cone. equal and their heights are equal, then their volumes are equal.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters c. If the base area is multiplied by 5, the volume is and a base area of 24 square centimeters. multiplied by 5. If the height is multiplied by 5, the b. VERBAL What is true about the volumes of the volume is multiplied by 5. If both the base area and two pyramids that you drew? Explain. the height are multiplied by 5, the volume is multiplied c. ANALYTICAL Explain how multiplying the base by 5 · 5 or 25. area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

36. CCSS ARGUMENTS Determine whether the SOLUTION: following statement is sometimes, always, or never The slant height is used for surface area, but the true. Justify your reasoning. height is used for volume. For this cone, the slant The volume of a cone with radius r and height h height of 13 is provided, and we need to calculate the equals the volume of a prism with height h. height before we can calculate the volume.

SOLUTION:

The volume of a cone with a radius r and height h is Alexandra incorrectly used the slant height. . The volume of a prism with a height of REASONING h is where B is the area of the base of the 38. A cone has a volume of 568 cubic prism. Set the volumes equal. centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning.

The volumes will only be equal when the radius of SOLUTION: the cone is equal to or when . The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of Therefore, the statement is true sometimes if the that. So, if a pyramid and prism have the same base, base area of the cone is 3 times as great as the base then in order to have the same volume, the height of area of the prism. For example, if the base of the the pyramid must be 3 times as great as the height of prism has an area of 10 square units, then its volume the prism. is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer. Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: The slant height is used for surface area, but the SOLUTION: height is used for volume. For this cone, the slant To find the volume of each solid, you must know the height of 13 is provided, and we need to calculate the area of the base and the height. The volume of a height before we can calculate the volume. pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the Alexandra incorrectly used the slant height. same height and base area.

38. REASONING A cone has a volume of 568 cubic 41. A conical sand toy has the dimensions as shown centimeters. What is the volume of a cylinder that below. How many cubic centimeters of sand will it has the same radius and height as the cone? Explain hold when it is filled to the top? your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone A 12π is V = Bh. The volume of a cylinder is three times B 15π as much as the volume of a cone with the same radius and height. C

39. OPEN ENDED Give an example of a pyramid and D a prism that have the same base and the same

volume. Explain your reasoning. SOLUTION: SOLUTION: Use the Pythagorean Theorem to find the radius r of The formula for volume of a prism is V = Bh and the the cone. formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the

height of the prism. So, the radius of the cone is 3 centimeters. Sample answer: A square pyramid with a base area of 16 and a The volume of a circular cone is , where height of 12, a prism with a square base area of 16 and a height of 4. r is the radius of the base and h is the height of the cone. 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a Therefore, the correct choice is A. cone is one third the volume of a cylinder that has the same height and base area. 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 41. A conical sand toy has the dimensions as shown 6 feet by 8 feet. If the tent holds 88 cubic feet of air, below. How many cubic centimeters of sand will it how tall is the tent’s center pole? hold when it is filled to the top? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

A 12π

B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone. 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

H So, the radius of the cone is 3 centimeters.

The volume of a circular cone is , where J

r is the radius of the base and h is the height of the SOLUTION: cone. Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

Therefore, the correct choice is A.

area of the base and h is the height of the pyramid. B x – 4

D E

SOLUTION:

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows So, the correct choice is E. the results of several spins. What is the experimental probability of the spinner landing on orange? Find the volume of each prism.

45.

SOLUTION: F The volume of a prism is , where B is the area of the base and h is the height of the prism. The G base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the H prism is 6 inches.

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 3 Favorable outcomes: {5 orange} Therefore, the volume of the prism is 1008 in . Number of favorable outcomes: 5

46. SOLUTION: The volume of a prism is , where B is the

So, the correct choice is F. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base 44. SAT/ACT For all of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to A –8 find the height of the triangle. B x – 4 C D E

SOLUTION:

So, the correct choice is E. So, the height of the triangle is 12 feet. Find the area Find the volume of each prism. of the triangle.

45. SOLUTION: The volume of a prism is , where B is the 2 area of the base and h is the height of the prism. The So, the area of the base B is 60 ft . base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the The height h of the prism is 19 feet. prism is 6 inches.

3 Therefore, the volume of the prism is 1140 ft .

3 Therefore, the volume of the prism is 1008 in .

47. SOLUTION: 46. The volume of a prism is , where B is the SOLUTION: area of the base and h is the height of the prism. The The volume of a prism is , where B is the base of this prism is a rectangle with a length of 79.4 area of the base and h is the height of the prism. The meters and a width of 52.5 meters. The height of the base of this prism is an isosceles triangle with a base prism is 102.3 meters. of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. So, the height of the triangle is 12 feet. Find the area Refer to the photo on Page 847. of the triangle. SOLUTION:

To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

2 So, the area of the base B is 60 ft . Find the slant height of the conical top.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1140 ft .

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) 47. or 9 ft.

Therefore, the volume of the prism is about 426,437.6 The formula for finding the surface area of a cylinder 3 is , where is r is the radius and h is the slant m . height of the cylinder.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the Surface area of the bin = Surface area of the dimensions in the diagram to find the entire surface cylinder + Surface area of the conical top + surface area of the bin with a conical top and bottom. Write area of the conical bottom. the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. Find the area of each shaded region. Polygons The formula for finding the surface area of a cone is in 50 - 52 are regular. , where is r is the radius and l is the slant height of the cone.

49. Find the slant height of the conical top. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the

The formula for finding the surface area of a cylinder interior angle is . The apothem bisects the is , where is r is the radius and h is the slant angle, so we use 30° when using trig to find the height of the cylinder. length of the apothem.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

51. SOLUTION: The shaded area is the area of the equilateral triangle A regular hexagon has 6 sides, so the measure of the less the area of the inscribed circle.

interior angle is . The apothem bisects the Find the area of the circle with a radius of 3.6 feet. angle, so we use 30° when using trig to find the length of the apothem.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three

Now find the area of the hexagon. Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 51. square feet.

SOLUTION: Subtract to find the area of shaded region. The shaded area is the area of the equilateral triangle

less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

Therefore, the area of the shaded region is about 2 26.6 ft .

So, the area of the circle is about 40.7 square feet.

The equilateral triangle can be divided into three circle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the Use a trigonometric ratio to find the value of b. length of the side of the equilateral triangle as shown below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle. Use trigonometric ratios to find the values of b and

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

Therefore, the area of the shaded region is about A(shaded region) = A(circumscribed circle) - A 2 26.6 ft . (equilateral triangle) + A(inscribed circle)

52. SOLUTION: Therefore, the area of the shaded region is about 2 The area of the shaded region is the area of the 168.2 mm . circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION: 3. a rectangular pyramid with a height of 5.2 meters The volume of a pyramid is , where B is the and a base 8 meters by 4.5 meters area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 The volume of a pyramid is , where B is the centimeters. The height of the pyramid is 12 area of the base and h is the height of the pyramid. centimeters. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

4. a square pyramid with a height of 14 meters and a 3. a rectangular pyramid with a height of 5.2 meters base with 8-meter side lengths and a base 8 meters by 4.5 meters SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 The base of this pyramid is a rectangle with a length meters. The height of the pyramid is 14 meters. of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

Find the volume of each cone. Round to the 4. a square pyramid with a height of 14 meters and a nearest tenth. base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the 5. area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 SOLUTION: meters. The height of the pyramid is 14 meters. The volume of a circular cone is , or

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

5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the 6. height of the cone, and r is the radius of the base. SOLUTION: Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

Use trigonometry to find the radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. 6. SOLUTION:

7. an oblique cone with a height of 10.5 millimeters and Use trigonometry to find the radius r. a radius of 1.6 millimeters

SOLUTION:

The volume of a circular cone is , or The volume of a circular cone is , or , where B is the area of the base, h is the , where B is the area of the base, h is the height of the cone, and r is the radius of the base. height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the The height of the cone is 11.5 centimeters. height is 10.5 millimeters.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

8. a cone with a slant height of 25 meters and a radius of 15 meters So, the height of the cone is 20 meters. SOLUTION:

Use the Pythagorean Theorem to find the height h of MUSEUMS the cone. Then find its volume. 9. The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B So, the height of the cone is 20 meters. is the area of the base and h is the height of the cone.

For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square 10. feet and the height is 100 feet. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. 11. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

11. 12. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

12-5 Volumes of Pyramids and Cones

12. 13. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The apothem is . The base is a hexagon, so we need to make a right tri

determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The apothem is .

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15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters 14. a pentagonal pyramid with a base area of 590 square SOLUTION: feet and an altitude of 7 feet Find the height of the right triangle. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a The volume of a pyramid is , where B is the of the right triangle and then find the area of the triangle. area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a The length of the other leg of the right triangle is 6 of the right triangle and then find the area of the cm. triangle.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The length of the other leg of the right triangle is 6 cm.

Therefore, the height of the triangular pyramid is 18 cm.

Find the volume of each cone. Round to the nearest tenth. So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. 17.

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the Therefore, the height of the triangular pyramid is 18 radius is or 5 inches. The height of the cone is 9 cm. inches.

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

17. 3 SOLUTION: Therefore, the volume of the cone is about 235.6 in .

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the 18. radius is or 5 inches. The height of the cone is 9 SOLUTION: inches. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 235.6 in3.

Therefore, the volume of the cone is about 134.8 3 cm . 18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the 19. cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. SOLUTION:

Therefore, the volume of the cone is about 134.8 Use a trigonometric ratio to find the height h of the 3 cm . cone.

19. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Use a trigonometric ratio to find the height h of the cone. Therefore, the volume of the cone is about 1473.1 3 cm .

The volume of a circular cone is , where 20. r is the radius of the base and h is the height of the SOLUTION: cone. The radius of this cone is 8 centimeters.

Use trigonometric ratios to find the height h and the radius r of the cone. Therefore, the volume of the cone is about 1473.1 cm3.

20. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone.

Use trigonometric ratios to find the height h and the

radius r of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION: The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 2.8 ft . Therefore, the volume of the cone is about 1072.3 21. an oblique cone with a diameter of 16 inches and an in 3. altitude of 16 inches 22. a right cone with a slant height of 5.6 centimeters SOLUTION: and a radius of 1 centimeter The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the The cone has a radius r of 1 centimeter and a slant cone. Since the diameter of this cone is 16 inches, height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 in 3. 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 5.8 cm .

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the volume of the cone is about 5.8 cm .

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest 3 Therefore, the paper cone will hold about 234.6 cm tenth. of roasted peanuts. SOLUTION: 24. CCSS MODELING The Pyramid Arena in The volume of a circular cone is , where Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its r is the radius of the base and h is the height of the square base is 600 feet wide. Find the volume of this cone. Since the diameter of the cone is 8 pyramid. centimeters, the radius is or 4 centimeters. The SOLUTION: height of the cone is 14 centimeters. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in 25. GARDENING The greenhouse is a regular Memphis, Tennessee, is the third largest pyramid in octagonal pyramid with a height of 5 feet. The base the world. It is approximately 350 feet tall, and its has side lengths of 2 feet. What is the volume of the square base is 600 feet wide. Find the volume of this greenhouse? pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is 25. GARDENING The greenhouse is a regular or 45°, so the angle formed in the triangle octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the below is 22.5°. greenhouse?

Use a trigonometric ratio to find the apothem a. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is The height of this pyramid is 5 feet. or 45°, so the angle formed in the triangle below is 22.5°.

Use a trigonometric ratio to find the apothem a. Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

The height of this pyramid is 5 feet.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

27.

SOLUTION:

28. 27. SOLUTION: SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase? The volume of the ceiling is

SOLUTION: The total volume is therefore 5000 + 1666.67 = 3 The building can be broken down into the rectangular 6666.67 ft . Two BTU's are needed for every cubic base and the pyramid ceiling. The volume of the base foot, so the size of the heating unit Sam should buy is is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation.

SOLUTION: The volume of the ceiling is The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. It tells Marta how much clay is needed to make the model.

30. SCIENCE Refer to page 825. Determine the 31. CHANGING DIMENSIONS A cone has a radius volume of the model. Explain why knowing the of 4 centimeters and a height of 9 centimeters. volume is helpful in this situation. Describe how each change affects the volume of the cone. SOLUTION: a. The height is doubled. b. The volume of a pyramid is , where B is the The radius is doubled. c. Both the radius and the height are doubled. area of the base and h is the height of the pyramid. SOLUTION: Find the volume of the original cone. Then alter the values.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. a. Double h. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION:

Find the volume of the original cone. Then alter the The volume is doubled.

values. b. Double r.

a. Double h. 2 The volume is multiplied by 2 or 4.

c. Double r and h.

The volume is doubled.

b. Double r. volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base.

2 The volume is multiplied by 2 or 4. SOLUTION:

The volume of a pyramid is , where B is the c. Double r and h. area of the base and h is the height of the pyramid. Let s be the side length of the base.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base.

SOLUTION:

The volume of a pyramid is , where B is the The side length of the base is 15 cm.

area of the base and h is the height of the pyramid. 33. The volume of a cone is 196π cubic inches and the

Let s be the side length of the base. height is 12 inches. What is the diameter?

SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

The diameter is 2(7) or 14 inches. , where B is the area of the base, h is the 34. The lateral area of a cone is 71.6 square millimeters height of the cone, and r is the radius of the base. and the slant height is 6 millimeters. What is the Since the diameter is 8 centimeters, the radius is 4 volume of the cone? centimeters. SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. The diameter is 2(7) or 14 inches. Replace L with 71.6 and with 6, then solve for the radius r. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

So, the radius is about 3.8 millimeters.

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

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the So, the radius is about 3.8 millimeters. cone.

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

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this So, the height of the cone is about 4.64 millimeters. problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters The volume of a circular cone is , where and a base area of 24 square centimeters. r is the radius of the base and h is the height of the b. VERBAL What is true about the volumes of the cone. two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Therefore, the volume of the cone is about 70.2 Sample answer: 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

b. The volumes are the same. The volume of a c. If the base area is multiplied by 5, the volume is pyramid equals one third times the base area times multiplied by 5. If the height is multiplied by 5, the the height. So, if the base areas of two pyramids are volume is multiplied by 5. If both the base area and equal and their heights are equal, then their volumes the height are multiplied by 5, the volume is multiplied are equal. by 5 · 5 or 25.

SOLUTION: The volume of a cone with a radius r and height h is The volumes will only be equal when the radius of . The volume of a prism with a height of the cone is equal to or when . h is where B is the area of the base of the Therefore, the statement is true sometimes if the prism. Set the volumes equal. base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base SOLUTION: area of 30 square units so that its volume is The slant height is used for surface area, but the or 10h cubic units. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume. ERROR ANALYSIS 37. Alexandra and Cornelio are calculating the volume of the cone below. Is either of

them correct? Explain your answer. Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain

your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone

is V = Bh. The volume of a cylinder is three times SOLUTION: as much as the volume of a cone with the same The slant height is used for surface area, but the radius and height. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the 39. OPEN ENDED Give an example of a pyramid and height before we can calculate the volume. a prism that have the same base and the same volume. Explain your reasoning.

Alexandra incorrectly used the slant height. SOLUTION: The formula for volume of a prism is V = Bh and the 38. REASONING A cone has a volume of 568 cubic formula for the volume of a pyramid is one-third of centimeters. What is the volume of a cylinder that that. So, if a pyramid and prism have the same base, has the same radius and height as the cone? Explain then in order to have the same volume, the height of your reasoning. the pyramid must be 3 times as great as the height of the prism. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

Set the base areas of the prism and pyramid, and 39. OPEN ENDED Give an example of a pyramid and make the height of the pyramid equal to 3 times the a prism that have the same base and the same height of the prism. volume. Explain your reasoning.

SOLUTION: Sample answer: The formula for volume of a prism is V = Bh and the A square pyramid with a base area of 16 and a formula for the volume of a pyramid is one-third of height of 12, a prism with a square base area of 16 that. So, if a pyramid and prism have the same base, and a height of 4. then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of 40. WRITING IN MATH Compare and contrast the prism. finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the Set the base areas of the prism and pyramid, and same height and base area. make the height of the pyramid equal to 3 times the height of the prism. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it Sample answer: hold when it is filled to the top? A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding A 12π volumes of prisms and cylinders. B 15π

SOLUTION: C To find the volume of each solid, you must know the

area of the base and the height. The volume of a D pyramid is one third the volume of a prism that has the same height and base area. The volume of a SOLUTION: cone is one third the volume of a cylinder that has the same height and base area. Use the Pythagorean Theorem to find the radius r of the cone. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π B 15π

SOLUTION: So, the radius of the cone is 3 centimeters. Use the Pythagorean Theorem to find the radius r of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, So, the radius of the cone is 3 centimeters. how tall is the tent’s center pole? The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the The volume of a pyramid is , where B is the cone. area of the base and h is the height of the pyramid.

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, 43. PROBABILITY A spinner has sections colored how tall is the tent’s center pole? red, blue, orange, and green. The table below shows the results of several spins. What is the experimental SOLUTION: probability of the spinner landing on orange? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

J 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows SOLUTION: the results of several spins. What is the experimental Possible outcomes: {6 red, 4 blue, 5 orange, 10 probability of the spinner landing on orange? green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

H So, the correct choice is F.

J 44. SAT/ACT For all

SOLUTION: A –8 Possible outcomes: {6 red, 4 blue, 5 orange, 10 B x – 4 green} C Number of possible outcomes : 25 Favorable outcomes: {5 orange} D Number of favorable outcomes: 5 E

SOLUTION:

So, the correct choice is F. So, the correct choice is E.

Find the volume of each prism. 44. SAT/ACT For all

A –8 B x – 4 C 45. D SOLUTION: E The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 SOLUTION: inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is E.

Find the volume of each prism.

3 Therefore, the volume of the prism is 1008 in .

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The 46. base of this prism is a rectangle with a length of 14 SOLUTION: inches and a width of 12 inches. The height h of the prism is 6 inches. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

3 Therefore, the volume of the prism is 1008 in .

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base So, the height of the triangle is 12 feet. Find the area of 10 feet and two legs of 13 feet. The height h will of the triangle. bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

So, the height of the triangle is 12 feet. Find the area 3 of the triangle. Therefore, the volume of the prism is 1140 ft .

47.

So, the area of the base B is 60 ft2. SOLUTION: The volume of a prism is , where B is the The height h of the prism is 19 feet. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4

meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

3 Therefore, the volume of the prism is 1140 ft .

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination 47. hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin SOLUTION: allows the grain to be emptied more easily. Use the The volume of a prism is , where B is the dimensions in the diagram to find the entire surface area of the base and h is the height of the prism. The area of the bin with a conical top and bottom. Write base of this prism is a rectangle with a length of 79.4 the exact answer and the answer rounded to the meters and a width of 52.5 meters. The height of the nearest square foot. prism is 102.3 meters. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is Therefore, the volume of the prism is about 426,437.6 , where is r is the radius and l is the slant height of the cone. m3.

SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. Find the slant height of the conical bottom. The formula for finding the surface area of a cone is The height of the conical bottom is 28 – (5 + 12 + 2) , where is r is the radius and l is the slant height or 9 ft.

of the cone.

Find the slant height of the conical top.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant

height of the cylinder.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular. The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder. 49. SOLUTION: Area of the shaded region = Area of the rectangle – Surface area of the bin = Surface area of the Area of the circle cylinder + Surface area of the conical top + surface Area of the rectangle = 10(5) area of the conical bottom. = 50

Find the area of each shaded region. Polygons in 50 - 52 are regular.

50. 49. SOLUTION: SOLUTION: Area of the shaded region = Area of the circle – Area of the shaded region = Area of the rectangle – Area of the hexagon Area of the circle Area of the rectangle = 10(5) = 50

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

51.

Now find the area of the hexagon. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. The equilateral triangle can be divided into three

Find the area of the circle with a radius of 3.6 feet. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

The equilateral triangle can be divided into three Now use the area of the isosceles triangles to find isosceles triangles with central angles of or the area of the equilateral triangle. 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet.

Use a trigonometric ratio to find the value of b. Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 Now use the area of the isosceles triangles to find 26.6 ft . the area of the equilateral triangle.

52. SOLUTION: The area of the shaded region is the area of the So, the area of the equilateral triangle is about 67.3 circumscribed circle minus the area of the equilateral square feet. triangle plus the area of the inscribed circle. Let b

represent the length of each side of the equilateral

Subtract to find the area of shaded region. triangle and h represent the radius of the inscribed circle.

Therefore, the area of the shaded region is about 2 26.6 ft .

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The 52. angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the SOLUTION: length of the side of the equilateral triangle as shown The area of the shaded region is the area of the below. circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Use trigonometric ratios to find the values of b and h.

Therefore, the area of the shaded region is about 168.2 mm2.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with 3. a rectangular pyramid with a height of 5.2 meters sides of 4.4 centimeters and an apothem of 3 and a base 8 meters by 4.5 meters centimeters. The height of the pyramid is 12 SOLUTION: centimeters. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters

SOLUTION: 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length The volume of a pyramid is , where B is the of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters. area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths Find the volume of each cone. Round to the SOLUTION: nearest tenth. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Find the volume of each cone. Round to the Since the diameter of this cone is 7 inches, the radius nearest tenth. is or 3.5 inches. The height of the cone is 4 inches.

5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius 6. is or 3.5 inches. The height of the cone is 4 inches. SOLUTION:

Use trigonometry to find the radius r.

The volume of a circular cone is , or

6. , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

Use trigonometry to find the radius r.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters The volume of a circular cone is , or SOLUTION:

, where B is the area of the base, h is the The volume of a circular cone is , or height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or 8. a cone with a slant height of 25 meters and a radius of 15 meters , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

So, the height of the cone is 20 meters.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. So, the height of the cone is 20 meters. SOLUTION: The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION: CCSS SENSE-MAKING Find the volume of The volume of a circular cone is , where B each pyramid. Round to the nearest tenth if necessary. is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION:

The volume of a pyramid is , where B is the 11. area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

11. SOLUTION:

The volume of a pyramid is , where B is the 12. area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

12. SOLUTION:

The volume of a pyramid is , where B is the 13. area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle. The apothem is .

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION: The apothem is . The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

12-5 Volumes of Pyramids and Cones

14. a pentagonal pyramid with a base area of 590 square 15. a triangular pyramid with a height of 4.8 centimeters feet and an altitude of 7 feet and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: SOLUTION: The volume of a pyramid is , where B is the Find the height of the right triangle. area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters

SOLUTION: Find the height of the right triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height.

SOLUTION: The volume of a pyramid is , where B is the The base of the pyramid is a right triangle with a leg area of the base and h is the height of the pyramid. of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

eSolutions Manual - Powered by Cognero Page 5 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The length of the other leg of the right triangle is 6 cm.

Therefore, the height of the triangular pyramid is 18 So, the area of the base B is 24 cm2. cm.

Replace V with 144 and B with 24 in the formula for Find the volume of each cone. Round to the the volume of a pyramid and solve for the height h. nearest tenth.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height Therefore, the height of the triangular pyramid is 18 of the cone. cm. Since the diameter of this cone is 10 inches, the Find the volume of each cone. Round to the nearest tenth. radius is or 5 inches. The height of the cone is 9 inches.

17. SOLUTION:

The volume of a circular cone is , 3 where r is the radius of the base and h is the height Therefore, the volume of the cone is about 235.6 in . of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches. 18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. Therefore, the volume of the cone is about 235.6 in3.

18. Therefore, the volume of the cone is about 134.8 SOLUTION: 3 cm . The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

19. SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm .

Use a trigonometric ratio to find the height h of the cone.

19. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Use a trigonometric ratio to find the height h of the cone.

Therefore, the volume of the cone is about 1473.1 cm3.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. 20.

SOLUTION:

Therefore, the volume of the cone is about 1473.1 3 cm . Use trigonometric ratios to find the height h and the radius r of the cone.

20. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use trigonometric ratios to find the height h and the radius r of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

The volume of a circular cone is , where 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches r is the radius of the base and h is the height of the cone. SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an

altitude of 16 inches SOLUTION: Therefore, the volume of the cone is about 1072.3 in 3. The volume of a circular cone is , where 22. a right cone with a slant height of 5.6 centimeters r is the radius of the base and h is the height of the and a radius of 1 centimeter cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches. SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter

SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 5.8 cm .

SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The 3 Therefore, the volume of the cone is about 5.8 cm . height of the cone is 14 centimeters.

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION: Therefore, the paper cone will hold about 234.6 cm3 The volume of a circular cone is , where of roasted peanuts. r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in centimeters, the radius is or 4 centimeters. The the world. It is approximately 350 feet tall, and its height of the cone is 14 centimeters. square base is 600 feet wide. Find the volume of this pyramid.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this 25. GARDENING The greenhouse is a regular pyramid. octagonal pyramid with a height of 5 feet. The base SOLUTION: has side lengths of 2 feet. What is the volume of the greenhouse? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION:

The volume of a pyramid is , where B is the 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base area of the base and h is the height of the pyramid. has side lengths of 2 feet. What is the volume of the The base of the pyramid is a regular octagon with greenhouse? sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

SOLUTION:

The volume of a pyramid is , where B is the Use a trigonometric ratio to find the apothem a. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle

below is 22.5°.

The height of this pyramid is 5 feet.

Use a trigonometric ratio to find the apothem a.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

The height of this pyramid is 5 feet. Find the volume of each solid. Round to the nearest tenth.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

27. SOLUTION:

27. SOLUTION: 28. SOLUTION:

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she 28. needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction SOLUTION: with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she The building can be broken down into the rectangular needs to determine the BTUs (British Thermal Units) base and the pyramid ceiling. The volume of the base required to heat the building. For new construction is with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

The volume of the ceiling is

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the The volume of the ceiling is volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The total volume is therefore 5000 + 1666.67 =

6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the It tells Marta how much clay is needed to make the volume of the model. Explain why knowing the model. volume is helpful in this situation. SOLUTION: 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. The volume of a pyramid is , where B is the Describe how each change affects the volume of the cone. area of the base and h is the height of the pyramid. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone.

a. The height is doubled. a. Double h. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

The volume is doubled.

b. Double r.

a. Double h.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

The volume is doubled.

b. Double r.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 2 The volume is multiplied by 2 or 4. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. c. Double r and h. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. Let s be the side length of the base. The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 The diameter is 2(7) or 14 inches. centimeters. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The diameter is 2(7) or 14 inches. The lateral area of a cone is , where r is 34. The lateral area of a cone is 71.6 square millimeters the radius and is the slant height of the cone. and the slant height is 6 millimeters. What is the Replace L with 71.6 and with 6, then solve for the volume of the cone? radius r. SOLUTION:

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone. The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

So, the height of the cone is about 4.64 millimeters.

So, the radius is about 3.8 millimeters. The volume of a circular cone is , where Use the Pythagorean Theorem to find the height of r is the radius of the base and h is the height of the the cone. cone.

Therefore, the volume of the cone is about 70.2 So, the height of the cone is about 4.64 millimeters. 3 mm . The volume of a circular cone is , where 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. r is the radius of the base and h is the height of the a. GEOMETRIC Draw two pyramids with

cone. different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: Therefore, the volume of the cone is about 70.2 3 a. Use rectangular bases and pick values that mm . multiply to make 24.

35. MULTIPLE REPRESENTATIONS In this Sample answer: problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal. The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer. The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units. SOLUTION: 37. ERROR ANALYSIS Alexandra and Cornelio are The slant height is used for surface area, but the calculating the volume of the cone below. Is either of height is used for volume. For this cone, the slant them correct? Explain your answer. height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: SOLUTION: 1704 cm3; The formula for the volume of a cylinder The slant height is used for surface area, but the is V= Bh, while the formula for the volume of a cone height is used for volume. For this cone, the slant is V = Bh. The volume of a cylinder is three times height of 13 is provided, and we need to calculate the height before we can calculate the volume. as much as the volume of a cone with the same

radius and height.

39. OPEN ENDED Give an example of a pyramid and Alexandra incorrectly used the slant height. a prism that have the same base and the same

38. REASONING A cone has a volume of 568 cubic volume. Explain your reasoning. centimeters. What is the volume of a cylinder that SOLUTION: has the same radius and height as the cone? Explain The formula for volume of a prism is V = Bh and the your reasoning. formula for the volume of a pyramid is one-third of SOLUTION: that. So, if a pyramid and prism have the same base, 3 then in order to have the same volume, the height of 1704 cm ; The formula for the volume of a cylinder the pyramid must be 3 times as great as the height of is V= Bh, while the formula for the volume of a cone the prism. is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. Set the base areas of the prism and pyramid, and SOLUTION: make the height of the pyramid equal to 3 times the The formula for volume of a prism is V = Bh and the height of the prism. formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, Sample answer: then in order to have the same volume, the height of A square pyramid with a base area of 16 and a the pyramid must be 3 times as great as the height of height of 12, a prism with a square base area of 16 the prism. and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the Set the base areas of the prism and pyramid, and area of the base and the height. The volume of a make the height of the pyramid equal to 3 times the pyramid is one third the volume of a prism that has height of the prism. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the Sample answer: same height and base area. A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 41. A conical sand toy has the dimensions as shown and a height of 4. below. How many cubic centimeters of sand will it hold when it is filled to the top? 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the A 12π area of the base and the height. The volume of a B 15π

pyramid is one third the volume of a prism that has C the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area. D

41. A conical sand toy has the dimensions as shown SOLUTION: below. How many cubic centimeters of sand will it Use the Pythagorean Theorem to find the radius r of hold when it is filled to the top? the cone.

A 12π B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the correct choice is A.

So, the radius of the cone is 3 centimeters. 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is The volume of a circular cone is , where 6 feet by 8 feet. If the tent holds 88 cubic feet of air, r is the radius of the base and h is the height of the how tall is the tent’s center pole? cone. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION: 43. PROBABILITY A spinner has sections colored The volume of a pyramid is , where B is the red, blue, orange, and green. The table below shows the results of several spins. What is the experimental area of the base and h is the height of the pyramid. probability of the spinner landing on orange?

43. PROBABILITY A spinner has sections colored H red, blue, orange, and green. The table below shows the results of several spins. What is the experimental J probability of the spinner landing on orange? SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5 F

J So, the correct choice is F. SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} 44. SAT/ACT For all Number of possible outcomes : 25 Favorable outcomes: {5 orange} A –8 Number of favorable outcomes: 5 B x – 4 C

D E

SOLUTION:

So, the correct choice is F.

44. SAT/ACT For all So, the correct choice is E. A –8 B x – 4 Find the volume of each prism. C D

E 45.

SOLUTION: SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches. So, the correct choice is E.

Find the volume of each prism.

45. 3 Therefore, the volume of the prism is 1008 in . SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches. 46.

SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to 3 Therefore, the volume of the prism is 1008 in . find the height of the triangle.

46.

SOLUTION:

The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle. So, the height of the triangle is 12 feet. Find the area of the triangle.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

So, the height of the triangle is 12 feet. Find the area of the triangle.

3 Therefore, the volume of the prism is 1140 ft .

So, the area of the base B is 60 ft2. 47. The height h of the prism is 19 feet. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters. 3 Therefore, the volume of the prism is 1140 ft .

Therefore, the volume of the prism is about 426,437.6 47. m3. SOLUTION: 48. FARMING The picture shows a combination The volume of a prism is , where B is the hopper cone and bin used by farmers to store grain area of the base and h is the height of the prism. The after harvest. The cone at the bottom of the bin base of this prism is a rectangle with a length of 79.4 allows the grain to be emptied more easily. Use the meters and a width of 52.5 meters. The height of the dimensions in the diagram to find the entire surface prism is 102.3 meters. area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the Therefore, the volume of the prism is about 426,437.6 surface area of the conical top and bottom and find m3. the surface area of the cylinder and add them. The formula for finding the surface area of a cone is 48. FARMING The picture shows a combination , where is r is the radius and l is the slant height hopper cone and bin used by farmers to store grain of the cone. after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the Find the slant height of the conical top. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. Find the slant height of the conical bottom. Find the slant height of the conical top. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) The formula for finding the surface area of a cylinder or 9 ft. is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant Find the area of each shaded region. Polygons height of the cylinder. in 50 - 52 are regular.

Surface area of the bin = Surface area of the 49. cylinder + Surface area of the conical top + surface SOLUTION:

area of the conical bottom. Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – 50.

Area of the circle SOLUTION: Area of the rectangle = 10(5) = 50 Area of the shaded region = Area of the circle –

Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem. 50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. 51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the

So, the area of the circle is about 40.7 square feet. height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the

height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle. Now use the area of the isosceles triangles to find

the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b. So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Now use the area of the isosceles triangles to find the area of the equilateral triangle. Therefore, the area of the shaded region is about 2 26.6 ft .

So, the area of the equilateral triangle is about 67.3 52. square feet. SOLUTION: Subtract to find the area of shaded region. The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle. Therefore, the area of the shaded region is about 2 26.6 ft .

52. Divide the equilateral triangle into three isosceles SOLUTION: triangles with central angles of or 120°. The The area of the shaded region is the area of the angle in the triangle formed by the height.h of the circumscribed circle minus the area of the equilateral isosceles triangle is 60° and the base is half the triangle plus the area of the inscribed circle. Let b length of the side of the equilateral triangle as shown represent the length of each side of the equilateral below. triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h. Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region. Use trigonometric ratios to find the values of b and h. A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

Find the volume of each pyramid. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

2. SOLUTION:

SOLUTION:

The volume of a circular cone is , or

and a base 8 meters by 4.5 meters

SOLUTION:

6. SOLUTION:

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths

SOLUTION: Use trigonometry to find the radius r. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

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

is or 3.5 inches. The height of the cone is 4 inches.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

Use trigonometry to find the radius r.

The volume of a circular cone is , or So, the height of the cone is 20 meters.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of 7. an oblique cone with a height of 10.5 millimeters and air that the heating and cooling systems would have a radius of 1.6 millimeters to accommodate. Round to the nearest tenth. SOLUTION: SOLUTION:

CCSS SENSE-MAKING Find the volume of 8. a cone with a slant height of 25 meters and a radius each pyramid. Round to the nearest tenth if of 15 meters necessary. SOLUTION:

10. SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

So, the height of the cone is 20 meters.

11.

to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. 13.

SOLUTION: The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square area of the base and h is the height of the pyramid. feet and an altitude of 7 feet

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

12-5 Volumes of Pyramids and Cones Therefore, the height of the triangular pyramid is 18 cm.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 3 cm. Therefore, the volume of the cone is about 235.6 in .

the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 Therefore, the volume of the cone is about 134.8 3 cm. cm .

Find the volume of each cone. Round to the eSolutionsnearestManual tenth.- Powered by Cognero Page 6

19. 17. SOLUTION: SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 Use a trigonometric ratio to find the height h of the

inches. cone.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the 3 cone. The radius of this cone is 8 centimeters. Therefore, the volume of the cone is about 235.6 in .

18.

20. SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm . Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1473.1 cm3.

Therefore, the volume of the cone is about 1072.3 20. in 3.

Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter 3 SOLUTION: Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean 24. CCSS MODELING The Pyramid Arena in Theorem to find the height h of the cone. Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

3 Therefore, the volume of the cone is about 5.8 cm .

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest SOLUTION: tenth. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a circular cone is , where The base of the pyramid is a regular octagon with r is the radius of the base and h is the height of the sides of 2 feet. A central angle of the octagon is cone. Since the diameter of the cone is 8 or 45°, so the angle formed in the triangle below is 22.5°. centimeters, the radius is or 4 centimeters. The

height of the cone is 14 centimeters.

Use a trigonometric ratio to find the apothem a. Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

25. GARDENING The greenhouse is a regular Find the volume of each solid. Round to the octagonal pyramid with a height of 5 feet. The base nearest tenth. has side lengths of 2 feet. What is the volume of the greenhouse?

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Use a trigonometric ratio to find the apothem a. 27.

SOLUTION:

The height of this pyramid is 5 feet.

28. Therefore, the volume of the greenhouse is about SOLUTION: 3 32.2 ft .

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

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

27.

SOLUTION: The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

SCIENCE 28. 30. Refer to page 825. Determine the volume of the model. Explain why knowing the SOLUTION: volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume of the ceiling is a. Double h.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation.

SOLUTION: 2 The volume of a pyramid is , where B is the The volume is multiplied by 2 or 4.

area of the base and h is the height of the pyramid. c. Double r and h.

It tells Marta how much clay is needed to make the 3 model. volume is multiplied by 2 or 8.

31. CHANGING DIMENSIONS A cone has a radius Find each measure. Round to the nearest tenth of 4 centimeters and a height of 9 centimeters. if necessary. Describe how each change affects the volume of the 32. A square pyramid has a volume of 862.5 cubic cone. centimeters and a height of 11.5 centimeters. Find the side length of the base. a. The height is doubled. b. The radius is doubled. SOLUTION: c. Both the radius and the height are doubled. The volume of a pyramid is , where B is the SOLUTION: Find the volume of the original cone. Then alter the area of the base and h is the height of the pyramid.

values. Let s be the side length of the base.

a. Double h.

The side length of the base is 15 cm.

The volume is doubled. 33. The volume of a cone is 196π cubic inches and the

height is 12 inches. What is the diameter? b. Double r. SOLUTION:

The volume of a circular cone is , or

c. Double r and h.

volume is multiplied by 23 or 8.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The side length of the base is 15 cm.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

Sample answer:

So, the radius is about 3.8 millimeters.

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

So, the height of the cone is about 4.64 millimeters.

b. The volumes are the same. The volume of a The volume of a circular cone is , where pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are r is the radius of the base and h is the height of the equal and their heights are equal, then their volumes cone. are equal.

Therefore, the volume of the cone is about 70.2 3 mm .

MULTIPLE REPRESENTATIONS 35. In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with c. If the base area is multiplied by 5, the volume is different bases that have a height of 10 centimeters multiplied by 5. If the height is multiplied by 5, the

and a base area of 24 square centimeters. volume is multiplied by 5. If both the base area and b. VERBAL What is true about the volumes of the the height are multiplied by 5, the volume is multiplied two pyramids that you drew? Explain. by 5 · 5 or 25. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

. The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

SOLUTION: 36. CCSS ARGUMENTS Determine whether the The slant height is used for surface area, but the following statement is sometimes, always, or never height is used for volume. For this cone, the slant true. Justify your reasoning. height of 13 is provided, and we need to calculate the The volume of a cone with radius r and height h height before we can calculate the volume. equals the volume of a prism with height h. SOLUTION: Alexandra incorrectly used the slant height. The volume of a cone with a radius r and height h is . The volume of a prism with a height of 38. REASONING A cone has a volume of 568 cubic h is where B is the area of the base of the centimeters. What is the volume of a cylinder that prism. Set the volumes equal. has the same radius and height as the cone? Explain your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning.

or 10h cubic units.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

D 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: Use the Pythagorean Theorem to find the radius r of SOLUTION: the cone. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. So, the radius of the cone is 3 centimeters.

A 12π B 15π

So, the radius of the cone is 3 centimeters. J The volume of a circular cone is , where r is the radius of the base and h is the height of the SOLUTION: cone. Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

Therefore, the correct choice is A.

The volume of a pyramid is , where B is the A –8 B area of the base and h is the height of the pyramid. x – 4 C D E

SOLUTION:

45. SOLUTION:

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

G base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches. H

46. SOLUTION: The volume of a prism is , where B is the So, the correct choice is F. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will 44. SAT/ACT For all bisect the base. Use the Pythagorean Theorem to find the height of the triangle. A –8 B x – 4 C D E

SOLUTION:

So, the correct choice is E. So, the height of the triangle is 12 feet. Find the area of the triangle. Find the volume of each prism.

45. SOLUTION: The volume of a prism is , where B is the So, the area of the base B is 60 ft2. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 The height h of the prism is 19 feet. inches and a width of 12 inches. The height h of the prism is 6 inches.

3 Therefore, the volume of the prism is 1140 ft .

3 Therefore, the volume of the prism is 1008 in .

Therefore, the volume of the prism is about 426,437.6 m3.

So, the area of the base B is 60 ft2. Find the slant height of the conical top.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1140 ft .

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 47.

The formula for finding the surface area of a cylinder Therefore, the volume of the prism is about 426,437.6 is , where is r is the radius and h is the slant 3 m . height of the cylinder.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons

in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50 Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the Find the area of the circle with a radius of 3.6 feet. angle, so we use 30° when using trig to find the length of the apothem.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three

Use a trigonometric ratio to find the value of b. Now find the area of the hexagon.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet. 51. SOLUTION: Subtract to find the area of shaded region.

The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 2 26.6 ft .

So, the area of the circle is about 40.7 square feet.

Now use the area of the isosceles triangles to find the area of the equilateral triangle. Use trigonometric ratios to find the values of b and h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A Therefore, the area of the shaded region is about (equilateral triangle) + A(inscribed circle) 2 26.6 ft .

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Find the volume of each pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

1. SOLUTION:

SOLUTION:

2. 4. a square pyramid with a height of 14 meters and a SOLUTION: base with 8-meter side lengths The volume of a pyramid is , where B is the SOLUTION:

area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 area of the base and h is the height of the pyramid. centimeters. The height of the pyramid is 12 The base of this pyramid is a square with sides of 8 centimeters. meters. The height of the pyramid is 14 meters.

Find the volume of each cone. Round to the nearest tenth. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION: 5. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length The volume of a circular cone is , or of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

6. SOLUTION:

Find the volume of each cone. Round to the nearest tenth. Use trigonometry to find the radius r.

The volume of a circular cone is , or 5. SOLUTION: , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The volume of a circular cone is , or The height of the cone is 11.5 centimeters.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

SOLUTION:

8. a cone with a slant height of 25 meters and a radius Use trigonometry to find the radius r. of 15 meters

SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the

height of the cone, and r is the radius of the base. Use the Pythagorean Theorem to find the height h of The height of the cone is 11.5 centimeters. the cone. Then find its volume.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters So, the height of the cone is 20 meters. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

8. a cone with a slant height of 25 meters and a radius The volume of a circular cone is , where B of 15 meters is the area of the base and h is the height of the

SOLUTION: cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

So, the height of the cone is 20 meters.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square 11. feet and the height is 100 feet. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. 12. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

11. 13. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

14. a pentagonal pyramid with a base area of 590 square 13. feet and an altitude of 7 feet SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has SOLUTION: a volume of 144 cubic centimeters. Find the height. Find the height of the right triangle. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

The length of the other leg of the right triangle is 6 cm. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

So, the area of the base B is 24 cm2.

16. A triangular pyramid with a right triangle base with a Replace V with 144 and B with 24 in the formula for leg 8 centimeters and hypotenuse 10 centimeters has the volume of a pyramid and solve for the height h. a volume of 144 cubic centimeters. Find the height.

Therefore, the height of the triangular pyramid is 18 cm.

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

17. SOLUTION:

The volume of a circular cone is , The length of the other leg of the right triangle is 6 where r is the radius of the base and h is the height cm. of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

3 Therefore, the volume of the cone is about 235.6 in .

18.

SOLUTION: Therefore, the height of the triangular pyramid is 18

cm. The volume of a circular cone is , where Find the volume of each cone. Round to the r is the radius of the base and h is the height of the nearest tenth. cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height Therefore, the volume of the cone is about 134.8 3 of the cone. cm .

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

19. SOLUTION:

Therefore, the volume of the cone is about 235.6 in3.

Use a trigonometric ratio to find the height h of the cone.

18. SOLUTION:

The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and r is the radius of the base and h is the height of the the height is 7.3 centimeters. cone. The radius of this cone is 8 centimeters.

12-5Therefore, Volumes ofthe Pyramids volume of andthe cone Cones is about 134.8 Therefore, the volume of the cone is about 1473.1 3 3 cm . cm .

20. 19. SOLUTION: SOLUTION:

Use trigonometric ratios to find the height h and the radius r of the cone. Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. cone.

Therefore, the volume of the cone is about 1473.1 3 cm3. Therefore, the volume of the cone is about 2.8 ft . 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

20. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Use trigonometric ratios to find the height h and the radius r of the cone. eSolutions Manual - Powered by Cognero Page 7

Therefore, the volume of the cone is about 1072.3 3 in . 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the The cone has a radius r of 1 centimeter and a slant cone. height of 5.6 centimeters. Use the Pythagorean

Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 5.8 cm .

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a Therefore, the volume of the cone is about 1072.3 paper cone that is 14 centimeters high and has a base 3 diameter of 8 centimeters? Round to the nearest in . tenth. 22. a right cone with a slant height of 5.6 centimeters SOLUTION: and a radius of 1 centimeter The volume of a circular cone is , where SOLUTION: The cone has a radius r of 1 centimeter and a slant r is the radius of the base and h is the height of the height of 5.6 centimeters. Use the Pythagorean cone. Since the diameter of the cone is 8 Theorem to find the height h of the cone. centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. 3 Therefore, the volume of the cone is about 5.8 cm . 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION: 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base The volume of a circular cone is , where has side lengths of 2 feet. What is the volume of the greenhouse? r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

SOLUTION:

The volume of a pyramid is , where B is the

3 area of the base and h is the height of the pyramid. Therefore, the paper cone will hold about 234.6 cm The base of the pyramid is a regular octagon with of roasted peanuts. sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in below is 22.5°. the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet. 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Therefore, the volume of the greenhouse is about 3 SOLUTION: 32.2 ft .

The volume of a pyramid is , where B is the Find the volume of each solid. Round to the nearest tenth. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°. 26.

SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet.

27. SOLUTION:

Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

26. SOLUTION: Volume of the solid given = Volume of the small 28. cone + Volume of the large cone SOLUTION:

27. SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

28. The volume of the ceiling is SOLUTION:

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she 30. SCIENCE Refer to page 825. Determine the needs to determine the BTUs (British Thermal Units) volume of the model. Explain why knowing the required to heat the building. For new construction volume is helpful in this situation. with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to SOLUTION: purchase? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base It tells Marta how much clay is needed to make the is model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. The volume of the ceiling is b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the a. Double h. volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume is doubled.

b. Double r.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius 2 of 4 centimeters and a height of 9 centimeters. The volume is multiplied by 2 or 4. Describe how each change affects the volume of the cone. c. Double r and h. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base.

a. Double h. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The volume is doubled.

b. Double r.

2 The volume is multiplied by 2 or 4. The side length of the base is 15 cm.

c. Double r and h. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the volume is multiplied by 23 or 8. height of the cone, and r is the radius of the base. Find each measure. Round to the nearest tenth Since the diameter is 8 centimeters, the radius is 4 if necessary. centimeters. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter?

SOLUTION: The lateral area of a cone is , where r is the radius and is the slant height of the cone. The volume of a circular cone is , or Replace L with 71.6 and with 6, then solve for the radius r. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the radius is about 3.8 millimeters.

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

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters So, the height of the cone is about 4.64 millimeters. and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The lateral area of a cone is , where r is Therefore, the volume of the cone is about 70.2 the radius and is the slant height of the cone. 3 Replace L with 71.6 and with 6, then solve for the mm . radius r. 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. So, the radius is about 3.8 millimeters. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the Use the Pythagorean Theorem to find the height of volume of the pyramid. the cone. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm . b. The volumes are the same. The volume of a pyramid equals one third times the base area times 35. MULTIPLE REPRESENTATIONS In this the height. So, if the base areas of two pyramids are problem, you will investigate rectangular pyramids. equal and their heights are equal, then their volumes a. GEOMETRIC Draw two pyramids with are equal. different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer: c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never c. If the base area is multiplied by 5, the volume is true. Justify your reasoning. multiplied by 5. If the height is multiplied by 5, the The volume of a cone with radius r and height h volume is multiplied by 5. If both the base area and equals the volume of a prism with height h. the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

The volumes will only be equal when the radius of

the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

36. CCSS ARGUMENTS Determine whether the 37. ERROR ANALYSIS Alexandra and Cornelio are following statement is sometimes, always, or never calculating the volume of the cone below. Is either of true. Justify your reasoning. them correct? Explain your answer. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is

. The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain The volumes will only be equal when the radius of your reasoning. the cone is equal to or when . SOLUTION: Therefore, the statement is true sometimes if the 3 1704 cm ; The formula for the volume of a cylinder base area of the cone is 3 times as great as the base is V= Bh, while the formula for the volume of a cone area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is V = Bh. The volume of a cylinder is three times is 10h cubic units. So, the cone must have a base as much as the volume of a cone with the same area of 30 square units so that its volume is radius and height.

or 10h cubic units. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same 37. ERROR ANALYSIS Alexandra and Cornelio are volume. Explain your reasoning. calculating the volume of the cone below. Is either of SOLUTION:

them correct? Explain your answer. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant Set the base areas of the prism and pyramid, and height of 13 is provided, and we need to calculate the make the height of the pyramid equal to 3 times the height before we can calculate the volume. height of the prism.

Sample answer: Alexandra incorrectly used the slant height. A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 38. REASONING A cone has a volume of 568 cubic and a height of 4. centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain 40. WRITING IN MATH Compare and contrast your reasoning. finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: 3 SOLUTION: 1704 cm ; The formula for the volume of a cylinder To find the volume of each solid, you must know the is V= Bh, while the formula for the volume of a cone area of the base and the height. The volume of a is V = Bh. The volume of a cylinder is three times pyramid is one third the volume of a prism that has as much as the volume of a cone with the same the same height and base area. The volume of a radius and height. cone is one third the volume of a cylinder that has the same height and base area. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same 41. A conical sand toy has the dimensions as shown volume. Explain your reasoning. below. How many cubic centimeters of sand will it hold when it is filled to the top? SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of A 12π the prism. B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of Set the base areas of the prism and pyramid, and the cone. make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a So, the radius of the cone is 3 centimeters. cone is one third the volume of a cylinder that has the same height and base area. The volume of a circular cone is , where

41. A conical sand toy has the dimensions as shown r is the radius of the base and h is the height of the

below. How many cubic centimeters of sand will it cone. hold when it is filled to the top?

A 12π B 15π Therefore, the correct choice is A. C 42. SHORT RESPONSE Brooke is buying a tent that

D is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION: Use the Pythagorean Theorem to find the radius r of SOLUTION: the cone. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows So, the radius of the cone is 3 centimeters. the results of several spins. What is the experimental probability of the spinner landing on orange? The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

H Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that J is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, SOLUTION: how tall is the tent’s center pole? Possible outcomes: {6 red, 4 blue, 5 orange, 10

SOLUTION: green} Number of possible outcomes : 25 The volume of a pyramid is , where B is the Favorable outcomes: {5 orange} Number of favorable outcomes: 5 area of the base and h is the height of the pyramid.

So, the correct choice is F.

44. SAT/ACT For all

43. PROBABILITY A spinner has sections colored A –8 red, blue, orange, and green. The table below shows B x – 4 the results of several spins. What is the experimental C probability of the spinner landing on orange? D E

SOLUTION:

H So, the correct choice is E. Find the volume of each prism. J

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} 45. Number of possible outcomes : 25 SOLUTION: Favorable outcomes: {5 orange} Number of favorable outcomes: 5 The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is F.

44. SAT/ACT For all 3 Therefore, the volume of the prism is 1008 in . A –8 B x – 4 C D E 46. SOLUTION: SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle. So, the correct choice is E.

Find the volume of each prism.

45.

SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the

prism is 6 inches. So, the height of the triangle is 12 feet. Find the area of the triangle.

3 Therefore, the volume of the prism is 1008 in . So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

46. SOLUTION: The volume of a prism is , where B is the 3 area of the base and h is the height of the prism. The Therefore, the volume of the prism is 1140 ft . base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

47. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

So, the height of the triangle is 12 feet. Find the area of the triangle. Therefore, the volume of the prism is about 426,437.6 m3.

Refer to the photo on Page 847.

SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find

the surface area of the cylinder and add them. 3 Therefore, the volume of the prism is 1140 ft . The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. Therefore, the volume of the prism is about 426,437.6 3 m . 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847.

SOLUTION: The formula for finding the surface area of a cylinder To find the entire surface area of the bin, find the is , where is r is the radius and h is the slant surface area of the conical top and bottom and find height of the cylinder. the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface Find the slant height of the conical top. area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) SOLUTION: or 9 ft. Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant 50. height of the cylinder. SOLUTION:

Area of the shaded region = Area of the circle – Area of the hexagon

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem. Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon Now find the area of the hexagon.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon. The equilateral triangle can be divided into three

Use a trigonometric ratio to find the value of b.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. Now use the area of the isosceles triangles to find the area of the equilateral triangle. Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet. So, the area of the equilateral triangle is about 67.3 Next, find the area of the equilateral triangle. square feet.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 The equilateral triangle can be divided into three 26.6 ft . isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle. 52.

SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral Use a trigonometric ratio to find the value of b. triangle and h represent the radius of the inscribed circle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle. Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below. So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Use trigonometric ratios to find the values of b and h. Therefore, the area of the shaded region is about 2 26.6 ft .

52.

SOLUTION: The area of the shaded region is the area of the The area of the equilateral triangle will equal three circumscribed circle minus the area of the equilateral times the area of any of the isosceles triangles. Find triangle plus the area of the inscribed circle. Let b the area of the shaded region. represent the length of each side of the equilateral triangle and h represent the radius of the inscribed A(shaded region) = A(circumscribed circle) - A circle. (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about Divide the equilateral triangle into three isosceles 168.2 mm2. triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters Find the volume of each pyramid. and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the 1. area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length SOLUTION: of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

2. SOLUTION:

The volume of a pyramid is , where B is the Find the volume of each cone. Round to the nearest tenth. area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

5. SOLUTION:

The volume of a circular cone is , or

6. SOLUTION:

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

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

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters

5. SOLUTION: SOLUTION: The volume of a circular cone is , or

The volume of a circular cone is , or , where B is the area of the base, h is the

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height of the cone, and r is the radius of the base. height is 10.5 millimeters. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

6. SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

Use trigonometry to find the radius r.

So, the height of the cone is 20 meters. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have 7. an oblique cone with a height of 10.5 millimeters and to accommodate. Round to the nearest tenth. a radius of 1.6 millimeters SOLUTION: SOLUTION: The volume of a circular cone is , where B The volume of a circular cone is , or is the area of the base and h is the height of the , where B is the area of the base, h is the cone. For this cone, the area of the base is 15,400 square height of the cone, and r is the radius of the base. feet and the height is 100 feet. The radius of this cone is 1.6 millimeters and the

height is 10.5 millimeters.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if 8. a cone with a slant height of 25 meters and a radius necessary. of 15 meters SOLUTION:

10.

SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the height of the cone is 20 meters.

11. SOLUTION: 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical The volume of a pyramid is , where B is the building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of area of the base and h is the height of the pyramid. air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters 13. and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: SOLUTION: The volume of a pyramid is , where B is the Find the height of the right triangle. base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet 16. A triangular pyramid with a right triangle base with a SOLUTION: leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters.

Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the height of the triangular pyramid is 18 cm.

Find the volume of each cone. Round to the 16. A triangular pyramid with a right triangle base with a nearest tenth. leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. 17. Use the Pythagorean Theorem to find the other leg a SOLUTION: of the right triangle and then find the area of the triangle. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 Therefore, the volume of the cone is about 235.6 in3. cm.

18.

SOLUTION: So, the area of the base B is 24 cm2. The volume of a circular cone is , where

Replace V with 144 and B with 24 in the formula for r is the radius of the base and h is the height of the the volume of a pyramid and solve for the height h. cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 134.8 3 Therefore, the height of the triangular pyramid is 18 cm . cm.

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

19.

17. SOLUTION: SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the Use a trigonometric ratio to find the height h of the radius is or 5 inches. The height of the cone is 9 cone. inches.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. Therefore, the volume of the cone is about 235.6 in3.

18. Therefore, the volume of the cone is about 1473.1 SOLUTION: cm3. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. 20.

SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm . Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

The volume of a circular cone is , where 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1473.1 cm3.

Therefore, the volume of the cone is about 1072.3 in 3. 20. SOLUTION: 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

12-5 Volumes of Pyramids and Cones 3 3 Therefore, the volume of the cone is about 2.8 ft . Therefore, the volume of the cone is about 5.8 cm .

21. an oblique cone with a diameter of 16 inches and an 23. SNACKS Approximately how many cubic altitude of 16 inches centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base SOLUTION: diameter of 8 centimeters? Round to the nearest The volume of a circular cone is , where tenth. r is the radius of the base and h is the height of the SOLUTION: cone. Since the diameter of this cone is 16 inches, The volume of a circular cone is , where the radius is or 8 inches. r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8

centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Therefore, the volume of the cone is about 1072.3 in 3.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

GARDENING 25. The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

3 Therefore, the volume of the cone is about 5.8 cm . eSolutions23. SNACKSManual -ApproximatelyPowered by Cognero how many cubic Page 8 centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest SOLUTION: tenth. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a circular cone is , where The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is r is the radius of the base and h is the height of the or 45°, so the angle formed in the triangle cone. Since the diameter of the cone is 8 below is 22.5°. centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Use a trigonometric ratio to find the apothem a.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its The height of this pyramid is 5 feet. square base is 600 feet wide. Find the volume of this

pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

SOLUTION:

Use a trigonometric ratio to find the apothem a. 27. SOLUTION:

The height of this pyramid is 5 feet.

28.

Therefore, the volume of the greenhouse is about SOLUTION: 3 32.2 ft .

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

26. 29. HEATING Sam is building an art studio in her SOLUTION: backyard. To buy a heating unit for the space, she Volume of the solid given = Volume of the small needs to determine the BTUs (British Thermal Units) cone + Volume of the large cone required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

27.

SOLUTION: The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the 28. volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

required to heat the building. For new construction model. with good insulation, there should be 2 BTUs per 31. CHANGING DIMENSIONS A cone has a radius cubic foot. What size unit does Sam need to of 4 centimeters and a height of 9 centimeters. purchase? Describe how each change affects the volume of the

cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION:

Find the volume of the original cone. Then alter the SOLUTION: values. The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

a. The volume of the ceiling is Double h.

The total volume is therefore 5000 + 1666.67 = The volume is doubled. 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is b. Double r. 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION: 2 The volume is multiplied by 2 or 4. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. c. Double r and h.

It tells Marta how much clay is needed to make the volume is multiplied by 23 or 8. model. Find each measure. Round to the nearest tenth 31. CHANGING DIMENSIONS A cone has a radius if necessary. of 4 centimeters and a height of 9 centimeters. 32. A square pyramid has a volume of 862.5 cubic Describe how each change affects the volume of the centimeters and a height of 11.5 centimeters. Find cone. the side length of the base. a. The height is doubled. SOLUTION: b. The radius is doubled. c. Both the radius and the height are doubled. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. Find the volume of the original cone. Then alter the Let s be the side length of the base. values.

a. Double h.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the The volume is doubled. height is 12 inches. What is the diameter?

SOLUTION: b. Double r. The volume of a circular cone is , or

c. Double r and h.

volume is multiplied by 23 or 8.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The side length of the base is 15 cm. So, the radius is about 3.8 millimeters. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? Use the Pythagorean Theorem to find the height of the cone. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters. So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters Therefore, the volume of the cone is about 70.2 3 and the slant height is 6 millimeters. What is the mm . volume of the cone? 35. MULTIPLE REPRESENTATIONS In this SOLUTION: problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid.

The lateral area of a cone is , where r is SOLUTION: the radius and is the slant height of the cone. a. Use rectangular bases and pick values that Replace L with 71.6 and with 6, then solve for the multiply to make 24. radius r. Sample answer:

So, the radius is about 3.8 millimeters.

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

So, the height of the cone is about 4.64 millimeters. b. The volumes are the same. The volume of a pyramid equals one third times the base area times The volume of a circular cone is , where the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes r is the radius of the base and h is the height of the are equal. cone.

Therefore, the volume of the cone is about 70.2 3 mm .

Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are 36. CCSS ARGUMENTS Determine whether the equal and their heights are equal, then their volumes following statement is sometimes, always, or never are equal. true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

The volume of a cone with radius r and height h height before we can calculate the volume.

equals the volume of a prism with height h.

SOLUTION: Alexandra incorrectly used the slant height. The volume of a cone with a radius r and height h is 38. REASONING A cone has a volume of 568 cubic . The volume of a prism with a height of centimeters. What is the volume of a cylinder that h is where B is the area of the base of the has the same radius and height as the cone? Explain prism. Set the volumes equal. your reasoning.

SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning.

SOLUTION: The volumes will only be equal when the radius of The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of the cone is equal to or when . that. So, if a pyramid and prism have the same base, Therefore, the statement is true sometimes if the then in order to have the same volume, the height of base area of the cone is 3 times as great as the base the pyramid must be 3 times as great as the height of area of the prism. For example, if the base of the the prism. prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4. 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders.

SOLUTION: SOLUTION: To find the volume of each solid, you must know the The slant height is used for surface area, but the area of the base and the height. The volume of a height is used for volume. For this cone, the slant pyramid is one third the volume of a prism that has height of 13 is provided, and we need to calculate the the same height and base area. The volume of a height before we can calculate the volume. cone is one third the volume of a cylinder that has the

same height and base area.

Alexandra incorrectly used the slant height. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it REASONING 38. A cone has a volume of 568 cubic hold when it is filled to the top? centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION:

3 1704 cm ; The formula for the volume of a cylinder A 12π is V= Bh, while the formula for the volume of a cone B 15π

is V = Bh. The volume of a cylinder is three times C as much as the volume of a cone with the same radius and height. D 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same SOLUTION: volume. Explain your reasoning. Use the Pythagorean Theorem to find the radius r of the cone. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. So, the radius of the cone is 3 centimeters.

Sample answer: The volume of a circular cone is , where A square pyramid with a base area of 16 and a r is the radius of the base and h is the height of the height of 12, a prism with a square base area of 16 cone. and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a Therefore, the correct choice is A. pyramid is one third the volume of a prism that has the same height and base area. The volume of a 42. SHORT RESPONSE Brooke is buying a tent that cone is one third the volume of a cylinder that has the is in the shape of a rectangular pyramid. The base is same height and base area. 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it SOLUTION: hold when it is filled to the top? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

A 12π B 15π

So, the radius of the cone is 3 centimeters. J The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the Possible outcomes: {6 red, 4 blue, 5 orange, 10 cone. green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is So, the correct choice is F. 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? 44. SAT/ACT For all SOLUTION: A –8 The volume of a pyramid is , where B is the B x – 4 area of the base and h is the height of the pyramid. C

D E

SOLUTION:

So, the correct choice is E. 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows Find the volume of each prism. the results of several spins. What is the experimental probability of the spinner landing on orange?

45. SOLUTION:

The volume of a prism is , where B is the F area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 G inches and a width of 12 inches. The height h of the prism is 6 inches.

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The So, the correct choice is F. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to 44. SAT/ACT For all find the height of the triangle. A –8 B x – 4 C D E

SOLUTION:

So, the height of the triangle is 12 feet. Find the area

So, the correct choice is E. of the triangle.

Find the volume of each prism.

45.

SOLUTION: So, the area of the base B is 60 ft2. The volume of a prism is , where B is the area of the base and h is the height of the prism. The The height h of the prism is 19 feet. base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

3 Therefore, the volume of the prism is 1140 ft .

3 Therefore, the volume of the prism is 1008 in .

47. SOLUTION: The volume of a prism is , where B is the 46. area of the base and h is the height of the prism. The SOLUTION: base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the The volume of a prism is , where B is the prism is 102.3 meters. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847.

So, the height of the triangle is 12 feet. Find the area SOLUTION: of the triangle. To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top. So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1140 ft .

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant Therefore, the volume of the prism is about 426,437.6 height of the cylinder. m3.

area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the Find the area of each shaded region. Polygons surface area of the conical top and bottom and find in 50 - 52 are regular. the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. 49.

Find the slant height of the conical top. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the The formula for finding the surface area of a cylinder length of the apothem. is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5)

= 50 Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon 51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

A regular hexagon has 6 sides, so the measure of the Find the area of the circle with a radius of 3.6 feet. interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

Now find the area of the hexagon.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet.

51. Subtract to find the area of shaded region. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 2 26.6 ft .

So, the area of the circle is about 40.7 square feet. 52. Next, find the area of the equilateral triangle. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle. Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the

length of the side of the equilateral triangle as shown below. Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find Use trigonometric ratios to find the values of b and the area of the equilateral triangle. h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A Therefore, the area of the shaded region is about (equilateral triangle) + A(inscribed circle) 2 26.6 ft .

52. Therefore, the area of the shaded region is about 168.2 mm2. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

Find the volume of each pyramid. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the 1. area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of The volume of a pyramid is , where B is the the pyramid is 5.2 meters. area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

2. SOLUTION:

SOLUTION:

The volume of a circular cone is , or

SOLUTION:

6. SOLUTION:

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

Use trigonometry to find the radius r. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

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

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

6. SOLUTION:

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

Use trigonometry to find the radius r.

The volume of a circular cone is , or So, the height of the cone is 20 meters.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of 7. an oblique cone with a height of 10.5 millimeters and air that the heating and cooling systems would have a radius of 1.6 millimeters to accommodate. Round to the nearest tenth. SOLUTION: SOLUTION: The volume of a circular cone is , or The volume of a circular cone is , where B

, where B is the area of the base, h is the is the area of the base and h is the height of the cone. height of the cone, and r is the radius of the base. For this cone, the area of the base is 15,400 square The radius of this cone is 1.6 millimeters and the feet and the height is 100 feet.

height is 10.5 millimeters.

CCSS SENSE-MAKING Find the volume of 8. a cone with a slant height of 25 meters and a radius each pyramid. Round to the nearest tenth if of 15 meters necessary. SOLUTION:

10.

Use the Pythagorean Theorem to find the height h of SOLUTION: the cone. Then find its volume. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

So, the height of the cone is 20 meters.

11. 9. MUSEUMS The sky dome of the National Corvette SOLUTION: Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the The volume of a pyramid is , where B is the base is about 15,400 square feet, find the volume of

air that the heating and cooling systems would have area of the base and h is the height of the pyramid. to accommodate. Round to the nearest tenth.

SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

12. SOLUTION:

The volume of a pyramid is , where B is the CCSS SENSE-MAKING Find the volume of area of the base and h is the height of the pyramid. each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square area of the base and h is the height of the pyramid. feet and an altitude of 7 feet

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and SOLUTION: hypotenuse 10.2 centimeters The volume of a pyramid is , where B is the SOLUTION: base and h is the height of the pyramid. Find the height of the right triangle.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet

SOLUTION: 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has The volume of a pyramid is , where B is the a volume of 144 cubic centimeters. Find the height.

area of the base and h is the height of the pyramid. SOLUTION:

The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the height of the triangular pyramid is 18 cm.

16. A triangular pyramid with a right triangle base with a Find the volume of each cone. Round to the leg 8 centimeters and hypotenuse 10 centimeters has nearest tenth. a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a 17. of the right triangle and then find the area of the SOLUTION: triangle. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm. Therefore, the volume of the cone is about 235.6 in3.

18.

So, the area of the base B is 24 cm2. SOLUTION: The volume of a circular cone is , where Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 Therefore, the volume of the cone is about 134.8 3 cm. cm . Find the volume of each cone. Round to the nearest tenth.

19. 17. SOLUTION: SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the

radius is or 5 inches. The height of the cone is 9 Use a trigonometric ratio to find the height h of the inches. cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 235.6 in3. cone. The radius of this cone is 8 centimeters.

18.

SOLUTION: Therefore, the volume of the cone is about 1473.1 3 The volume of a circular cone is , where cm . r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

20. SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm . Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft . The volume of a circular cone is , where 21. an oblique cone with a diameter of 16 inches and an r is the radius of the base and h is the height of the altitude of 16 inches cone. The radius of this cone is 8 centimeters. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1473.1 cm3.

Therefore, the volume of the cone is about 1072.3 3 20. in . SOLUTION: 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft . 3 Therefore, the volume of the cone is about 5.8 cm . 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a SOLUTION: paper cone that is 14 centimeters high and has a base The volume of a circular cone is , where diameter of 8 centimeters? Round to the nearest tenth. r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, SOLUTION: the radius is or 8 inches. The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter 3 SOLUTION: Therefore, the paper cone will hold about 234.6 cm The cone has a radius r of 1 centimeter and a slant of roasted peanuts. height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

3 Therefore, the volume of the cone is about 5.8 cm .

The volume of a circular cone is , where area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with r is the radius of the base and h is the height of the sides of 2 feet. A central angle of the octagon is cone. Since the diameter of the cone is 8 or 45°, so the angle formed in the triangle centimeters, the radius is or 4 centimeters. The below is 22.5°. height of the cone is 14 centimeters.

3 Use a trigonometric ratio to find the apothem a. Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

12-5 Volumes of Pyramids and Cones Therefore, the volume of the greenhouse is about 3 32.2 ft .

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Use a trigonometric ratio to find the apothem a. 27. SOLUTION:

The height of this pyramid is 5 feet.

28. Therefore, the volume of the greenhouse is about 3 SOLUTION: 32.2 ft .

Find the volume of each solid. Round to the eSolutionsnearestManual tenth.- Powered by Cognero Page 9

26.

SOLUTION: 29. HEATING Sam is building an art studio in her Volume of the solid given = Volume of the small backyard. To buy a heating unit for the space, she cone + Volume of the large cone needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

27.

SOLUTION: The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

28. 30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the SOLUTION: volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: SOLUTION: Find the volume of the original cone. Then alter the The building can be broken down into the rectangular values. base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is a. Double h.

The total volume is therefore 5000 + 1666.67 = 3 The volume is doubled. 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. b. Double r.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

2 The volume of a pyramid is , where B is the The volume is multiplied by 2 or 4.

area of the base and h is the height of the pyramid. c. Double r and h.

It tells Marta how much clay is needed to make the model. volume is multiplied by 23 or 8.

31. CHANGING DIMENSIONS A cone has a radius Find each measure. Round to the nearest tenth of 4 centimeters and a height of 9 centimeters. if necessary. Describe how each change affects the volume of the 32. A square pyramid has a volume of 862.5 cubic cone. centimeters and a height of 11.5 centimeters. Find a. The height is doubled. the side length of the base. b. The radius is doubled. SOLUTION: c. Both the radius and the height are doubled. The volume of a pyramid is , where B is the SOLUTION: Find the volume of the original cone. Then alter the area of the base and h is the height of the pyramid. values. Let s be the side length of the base.

a. Double h.

The side length of the base is 15 cm.

The volume is doubled. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? b. Double r. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 2 centimeters. The volume is multiplied by 2 or 4.

c. Double r and h.

volume is multiplied by 23 or 8.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The side length of the base is 15 cm.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the Therefore, the volume of the cone is about 70.2 3 volume of the cone? mm .

SOLUTION: 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. The lateral area of a cone is , where r is the radius and is the slant height of the cone. SOLUTION: Replace L with 71.6 and with 6, then solve for the a. Use rectangular bases and pick values that radius r. multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

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

So, the height of the cone is about 4.64 millimeters.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids.

a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters c. If the base area is multiplied by 5, the volume is and a base area of 24 square centimeters. multiplied by 5. If the height is multiplied by 5, the b. VERBAL What is true about the volumes of the volume is multiplied by 5. If both the base area and two pyramids that you drew? Explain. the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

. The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of

them correct? Explain your answer.

SOLUTION: 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never The slant height is used for surface area, but the true. Justify your reasoning. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the The volume of a cone with radius r and height h height before we can calculate the volume. equals the volume of a prism with height h.

SOLUTION: The volume of a cone with a radius r and height h is Alexandra incorrectly used the slant height. . The volume of a prism with a height of 38. REASONING A cone has a volume of 568 cubic h is where B is the area of the base of the centimeters. What is the volume of a cylinder that prism. Set the volumes equal. has the same radius and height as the cone? Explain your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning.

SOLUTION: The volumes will only be equal when the radius of The formula for volume of a prism is V = Bh and the the cone is equal to or when . formula for the volume of a pyramid is one-third of Therefore, the statement is true sometimes if the that. So, if a pyramid and prism have the same base, base area of the cone is 3 times as great as the base then in order to have the same volume, the height of area of the prism. For example, if the base of the the pyramid must be 3 times as great as the height of prism has an area of 10 square units, then its volume the prism. is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

cone is one third the volume of a cylinder that has the

same height and base area. Alexandra incorrectly used the slant height. 41. A conical sand toy has the dimensions as shown 38. REASONING A cone has a volume of 568 cubic below. How many cubic centimeters of sand will it centimeters. What is the volume of a cylinder that hold when it is filled to the top? has the same radius and height as the cone? Explain your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone A 12π is V = Bh. The volume of a cylinder is three times B 15π

as much as the volume of a cone with the same C radius and height.

39. OPEN ENDED Give an example of a pyramid and D a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: Use the Pythagorean Theorem to find the radius r of SOLUTION: the cone. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. So, the radius of the cone is 3 centimeters. Sample answer: A square pyramid with a base area of 16 and a The volume of a circular cone is , where height of 12, a prism with a square base area of 16 r is the radius of the base and h is the height of the and a height of 4. cone.

WRITING IN MATH 40. Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has Therefore, the correct choice is A. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the 42. SHORT RESPONSE Brooke is buying a tent that same height and base area. is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, 41. A conical sand toy has the dimensions as shown how tall is the tent’s center pole? below. How many cubic centimeters of sand will it hold when it is filled to the top? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

A 12π B 15π

H So, the radius of the cone is 3 centimeters. J The volume of a circular cone is , where r is the radius of the base and h is the height of the SOLUTION: cone. Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

Therefore, the correct choice is A.

SOLUTION: 44. SAT/ACT For all

The volume of a pyramid is , where B is the A –8 area of the base and h is the height of the pyramid. B x – 4 C

D E

SOLUTION:

PROBABILITY 43. A spinner has sections colored So, the correct choice is E. red, blue, orange, and green. The table below shows the results of several spins. What is the experimental Find the volume of each prism. probability of the spinner landing on orange?

45. SOLUTION:

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

G base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the

H prism is 6 inches.

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 3 Therefore, the volume of the prism is 1008 in . Favorable outcomes: {5 orange} Number of favorable outcomes: 5

SOLUTION:

So, the correct choice is E. So, the height of the triangle is 12 feet. Find the area of the triangle. Find the volume of each prism.

45.

SOLUTION: The volume of a prism is , where B is the So, the area of the base B is 60 ft2. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 The height h of the prism is 19 feet. inches and a width of 12 inches. The height h of the

prism is 6 inches.

3 Therefore, the volume of the prism is 1140 ft .

3 Therefore, the volume of the prism is 1008 in .

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot.

Refer to the photo on Page 847. So, the height of the triangle is 12 feet. Find the area of the triangle. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

So, the area of the base B is 60 ft2. Find the slant height of the conical top.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1140 ft .

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 47. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

The formula for finding the surface area of a cylinder Therefore, the volume of the prism is about 426,437.6 is , where is r is the radius and h is the slant m3. height of the cylinder.

the surface area of the cylinder and add them. in 50 - 52 are regular. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. 49. Find the slant height of the conical top. SOLUTION:

Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50 Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the Find the area of the circle with a radius of 3.6 feet. angle, so we use 30° when using trig to find the length of the apothem.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b. Now find the area of the hexagon.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet.

51. SOLUTION: Subtract to find the area of shaded region. The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 2 26.6 ft .

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. 52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle. The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the

isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown Use a trigonometric ratio to find the value of b. below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle. Use trigonometric ratios to find the values of b and h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

Therefore, the area of the shaded region is about A(shaded region) = A(circumscribed circle) - A 2 (equilateral triangle) + A(inscribed circle) 26.6 ft .

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION: 3. a rectangular pyramid with a height of 5.2 meters The volume of a pyramid is , where B is the and a base 8 meters by 4.5 meters area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 The volume of a pyramid is , where B is the centimeters. The height of the pyramid is 12 centimeters. area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

3. a rectangular pyramid with a height of 5.2 meters 4. a square pyramid with a height of 14 meters and a and a base 8 meters by 4.5 meters base with 8-meter side lengths SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length The base of this pyramid is a square with sides of 8 of 8 meters and a width of 4.5 meters. The height of meters. The height of the pyramid is 14 meters. the pyramid is 5.2 meters.

Find the volume of each cone. Round to the 4. a square pyramid with a height of 14 meters and a nearest tenth. base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. 5. The base of this pyramid is a square with sides of 8 SOLUTION: meters. The height of the pyramid is 14 meters. The volume of a circular cone is , or

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

5. SOLUTION:

The volume of a circular cone is , or

Use trigonometry to find the radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. 6. The height of the cone is 11.5 centimeters.

SOLUTION:

Use trigonometry to find the radius r. 7. an oblique cone with a height of 10.5 millimeters and

a radius of 1.6 millimeters SOLUTION:

7. an oblique cone with a height of 10.5 millimeters and

a radius of 1.6 millimeters 8. a cone with a slant height of 25 meters and a radius SOLUTION: of 15 meters SOLUTION: The volume of a circular cone is , or

8. a cone with a slant height of 25 meters and a radius of 15 meters So, the height of the cone is 20 meters. SOLUTION:

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

So, the height of the cone is 20 meters. The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

to accommodate. Round to the nearest tenth. each pyramid. Round to the nearest tenth if SOLUTION: necessary.

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square 10. feet and the height is 100 feet. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION: 11.

The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

11. 12. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

12. SOLUTION: 13. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri The apothem is . determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the The apothem is . area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION: Find the height of the right triangle. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. The volume of a pyramid is , where B is the Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the area of the base and h is the height of the pyramid. triangle.

Use the Pythagorean Theorem to find the other leg a The length of the other leg of the right triangle is 6 of the right triangle and then find the area of the triangle. cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The length of the other leg of the right triangle is 6 cm.

Therefore, the height of the triangular pyramid is 18 cm.

Find the volume of each cone. Round to the So, the area of the base B is 24 cm2. nearest tenth.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the Therefore, the height of the triangular pyramid is 18 radius is or 5 inches. The height of the cone is 9 cm. inches. Find the volume of each cone. Round to the nearest tenth.

17.

SOLUTION: Therefore, the volume of the cone is about 235.6 in3. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the 18. radius is or 5 inches. The height of the cone is 9 SOLUTION: inches. The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 235.6 in3.

Therefore, the volume of the cone is about 134.8 3 cm . 18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and 19. the height is 7.3 centimeters. SOLUTION:

Therefore, the volume of the cone is about 134.8 3 Use a trigonometric ratio to find the height h of the cm . cone.

19. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Use a trigonometric ratio to find the height h of the cone. Therefore, the volume of the cone is about 1473.1 cm3.

The volume of a circular cone is , where 20. r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. SOLUTION:

Use trigonometric ratios to find the height h and the Therefore, the volume of the cone is about 1473.1 radius r of the cone.

cm3.

20. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone.

Use trigonometric ratios to find the height h and the radius r of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

3 Therefore, the volume of the cone is about 2.8 ft . Therefore, the volume of the cone is about 1072.3 21. an oblique cone with a diameter of 16 inches and an 3 altitude of 16 inches in . SOLUTION: 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the The cone has a radius r of 1 centimeter and a slant cone. Since the diameter of this cone is 16 inches, height of 5.6 centimeters. Use the Pythagorean the radius is or 8 inches. Theorem to find the height h of the cone.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 5.8 cm .

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest 3 tenth. Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. SOLUTION: 24. CCSS MODELING The Pyramid Arena in The volume of a circular cone is , where Memphis, Tennessee, is the third largest pyramid in r is the radius of the base and h is the height of the the world. It is approximately 350 feet tall, and its cone. Since the diameter of the cone is 8 square base is 600 feet wide. Find the volume of this pyramid. centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with 25. GARDENING The greenhouse is a regular sides of 2 feet. A central angle of the octagon is octagonal pyramid with a height of 5 feet. The base or 45°, so the angle formed in the triangle has side lengths of 2 feet. What is the volume of the below is 22.5°. greenhouse?

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

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle The height of this pyramid is 5 feet. below is 22.5°.

Use a trigonometric ratio to find the apothem a. Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

The height of this pyramid is 5 feet.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

27. SOLUTION:

28. 27. SOLUTION: SOLUTION:

12-5 Volumes of Pyramids and Cones

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base 29. HEATING Sam is building an art studio in her is backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase? The volume of the ceiling is

SOLUTION: The total volume is therefore 5000 + 1666.67 = The building can be broken down into the rectangular 6666.67 ft3. Two BTU's are needed for every cubic base and the pyramid ceiling. The volume of the base foot, so the size of the heating unit Sam should buy is is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation.

The volume of the ceiling is SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic eSolutionsfoot,Manual so the- sizePowered of theby Cogneroheating unit Sam should buy is Page 10 6666.67 × 2 = 13,333 BTUs. It tells Marta how much clay is needed to make the model.

30. SCIENCE Refer to page 825. Determine the 31. CHANGING DIMENSIONS A cone has a radius volume of the model. Explain why knowing the of 4 centimeters and a height of 9 centimeters. volume is helpful in this situation. Describe how each change affects the volume of the SOLUTION: cone. a. The height is doubled. The volume of a pyramid is , where B is the b. The radius is doubled. c. area of the base and h is the height of the pyramid. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the a. Double h. cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values. The volume is doubled.

b. Double r.

a. Double h. 2 The volume is multiplied by 2 or 4.

c. Double r and h.

The volume is doubled.

b. Double r. volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. 2 The volume is multiplied by 2 or 4. SOLUTION:

c. Double r and h. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the The side length of the base is 15 cm. area of the base and h is the height of the pyramid. Let s be the side length of the base. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the The diameter is 2(7) or 14 inches. height of the cone, and r is the radius of the base. 34. The lateral area of a cone is 71.6 square millimeters Since the diameter is 8 centimeters, the radius is 4 and the slant height is 6 millimeters. What is the centimeters. volume of the cone?

SOLUTION:

So, the radius is about 3.8 millimeters.

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

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the So, the radius is about 3.8 millimeters. cone.

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

Therefore, the volume of the cone is about 70.2 3 mm .

So, the height of the cone is about 4.64 millimeters. 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with The volume of a circular cone is , where different bases that have a height of 10 centimeters and a base area of 24 square centimeters. r is the radius of the base and h is the height of the b. VERBAL What is true about the volumes of the cone. two pyramids that you drew? Explain.

c. ANALYTICAL Explain how multiplying the base

area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24. Therefore, the volume of the cone is about 70.2 3 Sample answer: mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times c. If the base area is multiplied by 5, the volume is the height. So, if the base areas of two pyramids are multiplied by 5. If the height is multiplied by 5, the equal and their heights are equal, then their volumes volume is multiplied by 5. If both the base area and are equal. the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

The volume of a cone with a radius r and height h is The volumes will only be equal when the radius of . The volume of a prism with a height of the cone is equal to or when . h is where B is the area of the base of the prism. Set the volumes equal. Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

The volumes will only be equal when the radius of the cone is equal to or when .

is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is SOLUTION: The slant height is used for surface area, but the or 10h cubic units. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the 37. ERROR ANALYSIS Alexandra and Cornelio are height before we can calculate the volume. calculating the volume of the cone below. Is either of them correct? Explain your answer. Alexandra incorrectly used the slant height.

height is used for volume. For this cone, the slant radius and height. height of 13 is provided, and we need to calculate the 39. OPEN ENDED Give an example of a pyramid and height before we can calculate the volume. a prism that have the same base and the same

volume. Explain your reasoning.

1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and Set the base areas of the prism and pyramid, and a prism that have the same base and the same make the height of the pyramid equal to 3 times the volume. Explain your reasoning. height of the prism.

SOLUTION: Sample answer: The formula for volume of a prism is V = Bh and the A square pyramid with a base area of 16 and a formula for the volume of a pyramid is one-third of height of 12, a prism with a square base area of 16 that. So, if a pyramid and prism have the same base, and a height of 4. then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of 40. WRITING IN MATH Compare and contrast the prism. finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a Set the base areas of the prism and pyramid, and cone is one third the volume of a cylinder that has the make the height of the pyramid equal to 3 times the same height and base area. height of the prism. 41. A conical sand toy has the dimensions as shown Sample answer: below. How many cubic centimeters of sand will it A square pyramid with a base area of 16 and a hold when it is filled to the top? height of 12, a prism with a square base area of 16 and a height of 4.

WRITING IN MATH 40. Compare and contrast finding volumes of pyramids and cones with finding A 12π volumes of prisms and cylinders. B 15π SOLUTION: C To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has D the same height and base area. The volume of a cone is one third the volume of a cylinder that has the SOLUTION: same height and base area. Use the Pythagorean Theorem to find the radius r of the cone. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of So, the radius of the cone is 3 centimeters. the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the correct choice is A.

r is the radius of the base and h is the height of the The volume of a pyramid is , where B is the cone. area of the base and h is the height of the pyramid.

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, PROBABILITY how tall is the tent’s center pole? 43. A spinner has sections colored red, blue, orange, and green. The table below shows SOLUTION: the results of several spins. What is the experimental probability of the spinner landing on orange? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

H So, the correct choice is F.

J 44. SAT/ACT For all SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 A –8 green} B x – 4 Number of possible outcomes : 25 C Favorable outcomes: {5 orange} Number of favorable outcomes: 5 D E

SOLUTION:

So, the correct choice is F. So, the correct choice is E.

44. SAT/ACT For all Find the volume of each prism.

A –8 B x – 4 C 45. D SOLUTION: E The volume of a prism is , where B is the area of the base and h is the height of the prism. The SOLUTION: base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is E.

Find the volume of each prism.

3 Therefore, the volume of the prism is 1008 in .

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The 46. base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the SOLUTION: prism is 6 inches. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

3 Therefore, the volume of the prism is 1008 in .

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

So, the height of the triangle is 12 feet. Find the area 3 of the triangle. Therefore, the volume of the prism is 1140 ft .

47.

3 Therefore, the volume of the prism is 1140 ft .

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination 47. hopper cone and bin used by farmers to store grain SOLUTION: after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the The volume of a prism is , where B is the dimensions in the diagram to find the entire surface area of the base and h is the height of the prism. The area of the bin with a conical top and bottom. Write base of this prism is a rectangle with a length of 79.4 the exact answer and the answer rounded to the meters and a width of 52.5 meters. The height of the nearest square foot. prism is 102.3 meters. Refer to the photo on Page 847.

SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is Therefore, the volume of the prism is about 426,437.6 , where is r is the radius and l is the slant height m3. of the cone.

48. FARMING The picture shows a combination Find the slant height of the conical top. hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is Find the slant height of the conical bottom. , where is r is the radius and l is the slant height The height of the conical bottom is 28 – (5 + 12 + 2) of the cone. or 9 ft.

Find the slant height of the conical top.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant Find the slant height of the conical bottom. height of the cylinder. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Area of the shaded region = Area of the rectangle – Surface area of the bin = Surface area of the Area of the circle cylinder + Surface area of the conical top + surface Area of the rectangle = 10(5)

area of the conical bottom. = 50

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. 50. SOLUTION: SOLUTION: Area of the shaded region = Area of the rectangle – Area of the shaded region = Area of the circle –

Area of the circle Area of the hexagon Area of the rectangle = 10(5) = 50

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

51. Now find the area of the hexagon. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. The equilateral triangle can be divided into three Find the area of the circle with a radius of 3.6 feet. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

The equilateral triangle can be divided into three isosceles triangles with central angles of or Now use the area of the isosceles triangles to find

120°, so the angle in the triangle created by the the area of the equilateral triangle. height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet. Use a trigonometric ratio to find the value of b. Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about Now use the area of the isosceles triangles to find 2 the area of the equilateral triangle. 26.6 ft .

52. SOLUTION: So, the area of the equilateral triangle is about 67.3 The area of the shaded region is the area of the square feet. circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b Subtract to find the area of shaded region. represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Therefore, the area of the shaded region is about 2 26.6 ft .

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The 52. angle in the triangle formed by the height.h of the SOLUTION: isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown The area of the shaded region is the area of the below. circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle) Use trigonometric ratios to find the values of b and h.

Therefore, the area of the shaded region is about 2 168.2 mm .

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid.

1. SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters 4. a square pyramid with a height of 14 meters and a and a base 8 meters by 4.5 meters base with 8-meter side lengths SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length The base of this pyramid is a square with sides of 8 of 8 meters and a width of 4.5 meters. The height of meters. The height of the pyramid is 14 meters. the pyramid is 5.2 meters.

Find the volume of each cone. Round to the 4. a square pyramid with a height of 14 meters and a nearest tenth. base with 8-meter side lengths SOLUTION:

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

5. SOLUTION:

The volume of a circular cone is , or

Use trigonometry to find the radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. 6.

SOLUTION:

7. an oblique cone with a height of 10.5 millimeters and Use trigonometry to find the radius r. a radius of 1.6 millimeters

SOLUTION:

8. a cone with a slant height of 25 meters and a radius

of 15 meters So, the height of the cone is 20 meters. SOLUTION:

The volume of a circular cone is , where B So, the height of the cone is 20 meters. is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square 10. feet and the height is 100 feet. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION: 11. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

11. 12. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri The apothem is . determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a The volume of a pyramid is , where B is the of the right triangle and then find the area of the area of the base and h is the height of the pyramid. triangle.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a The length of the other leg of the right triangle is 6 of the right triangle and then find the area of the cm. triangle.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The length of the other leg of the right triangle is 6 cm.

Therefore, the height of the triangular pyramid is 18 cm.

Find the volume of each cone. Round to the So, the area of the base B is 24 cm2. nearest tenth.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

17.

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

17. SOLUTION: Therefore, the volume of the cone is about 235.6 in3.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the 18. radius is or 5 inches. The height of the cone is 9 SOLUTION: inches. The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 235.6 in3.

Therefore, the volume of the cone is about 134.8 3 cm . 18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and 19. the height is 7.3 centimeters. SOLUTION:

Therefore, the volume of the cone is about 134.8 Use a trigonometric ratio to find the height h of the 3 cm . cone.

19. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Use a trigonometric ratio to find the height h of the cone. Therefore, the volume of the cone is about 1473.1 cm3.

The volume of a circular cone is , where 20. r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. SOLUTION:

Use trigonometric ratios to find the height h and the

radius r of the cone. Therefore, the volume of the cone is about 1473.1 cm3.

20. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone.

Use trigonometric ratios to find the height h and the radius r of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

3 Therefore, the volume of the cone is about 2.8 ft . Therefore, the volume of the cone is about 1072.3 21. an oblique cone with a diameter of 16 inches and an in 3. altitude of 16 inches SOLUTION: 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the The cone has a radius r of 1 centimeter and a slant cone. Since the diameter of this cone is 16 inches, height of 5.6 centimeters. Use the Pythagorean the radius is or 8 inches. Theorem to find the height h of the cone.

Therefore, the volume of the cone is about 1072.3 3 in . 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 5.8 cm .

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the volume of the cone is about 5.8 cm .

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest 3 tenth. Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. SOLUTION: 24. CCSS MODELING The Pyramid Arena in The volume of a circular cone is , where Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its r is the radius of the base and h is the height of the square base is 600 feet wide. Find the volume of this cone. Since the diameter of the cone is 8 pyramid. centimeters, the radius is or 4 centimeters. The SOLUTION: height of the cone is 14 centimeters. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base or 45°, so the angle formed in the triangle has side lengths of 2 feet. What is the volume of the below is 22.5°. greenhouse?

Use a trigonometric ratio to find the apothem a. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is The height of this pyramid is 5 feet. or 45°, so the angle formed in the triangle

below is 22.5°.

Use a trigonometric ratio to find the apothem a. Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

The height of this pyramid is 5 feet.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

27. SOLUTION:

28. 27. SOLUTION: SOLUTION:

28. purchase? SOLUTION:

cubic foot. What size unit does Sam need to purchase? The volume of the ceiling is

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation.

The volume of the ceiling is SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 12-56666.67 × 2 = 13,333 BTUs. Volumes of Pyramids and Cones It tells Marta how much clay is needed to make the model.

30. SCIENCE Refer to page 825. Determine the 31. CHANGING DIMENSIONS A cone has a radius volume of the model. Explain why knowing the of 4 centimeters and a height of 9 centimeters. volume is helpful in this situation. Describe how each change affects the volume of the cone. SOLUTION: a. The height is doubled. The volume of a pyramid is , where B is the b. The radius is doubled. c. Both the radius and the height are doubled. area of the base and h is the height of the pyramid. SOLUTION: Find the volume of the original cone. Then alter the values.

It tells Marta how much clay is needed to make the model.

values. The volume is doubled.

b. Double r.

a. Double h. 2 The volume is multiplied by 2 or 4.

c. Double r and h.

The volume is doubled.

b. Double r. volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth eSolutions Manual - Powered by Cognero if necessary. Page 11 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. 2 The volume is multiplied by 2 or 4. SOLUTION:

c. Double r and h. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

The diameter is 2(7) or 14 inches. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. 34. The lateral area of a cone is 71.6 square millimeters Since the diameter is 8 centimeters, the radius is 4 and the slant height is 6 millimeters. What is the centimeters. volume of the cone? SOLUTION:

So, the radius is about 3.8 millimeters.

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

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the

radius r.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the So, the radius is about 3.8 millimeters. cone.

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

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this So, the height of the cone is about 4.64 millimeters. problem, you will investigate rectangular pyramids.

a. GEOMETRIC Draw two pyramids with The volume of a circular cone is , where different bases that have a height of 10 centimeters and a base area of 24 square centimeters. r is the radius of the base and h is the height of the b. VERBAL What is true about the volumes of the cone. two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Therefore, the volume of the cone is about 70.2 3 Sample answer: mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

prism. Set the volumes equal. base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is SOLUTION: The slant height is used for surface area, but the or 10h cubic units. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the

37. ERROR ANALYSIS Alexandra and Cornelio are height before we can calculate the volume.

calculating the volume of the cone below. Is either of

them correct? Explain your answer. Alexandra incorrectly used the slant height.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding A 12π volumes of prisms and cylinders. B 15π

SOLUTION: C To find the volume of each solid, you must know the

area of the base and the height. The volume of a D pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the SOLUTION: same height and base area. Use the Pythagorean Theorem to find the radius r of the cone. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π B 15π

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, 43. PROBABILITY A spinner has sections colored how tall is the tent’s center pole? red, blue, orange, and green. The table below shows SOLUTION: the results of several spins. What is the experimental probability of the spinner landing on orange? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

H So, the correct choice is F.

J 44. SAT/ACT For all SOLUTION: A –8 Possible outcomes: {6 red, 4 blue, 5 orange, 10 B x – 4 green} Number of possible outcomes : 25 C

Favorable outcomes: {5 orange} D Number of favorable outcomes: 5 E

SOLUTION:

So, the correct choice is F. So, the correct choice is E.

Find the volume of each prism. 44. SAT/ACT For all

So, the correct choice is E.

Find the volume of each prism.

3 Therefore, the volume of the prism is 1008 in .

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

So, the height of the triangle is 12 feet. Find the area 3 of the triangle. Therefore, the volume of the prism is 1140 ft .

47.

3 Therefore, the volume of the prism is 1140 ft .

Therefore, the volume of the prism is about 426,437.6 m3.

SOLUTION:

To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is Therefore, the volume of the prism is about 426,437.6 , where is r is the radius and l is the slant height m3. of the cone.

SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is Find the slant height of the conical bottom. , where is r is the radius and l is the slant height The height of the conical bottom is 28 – (5 + 12 + 2)

of the cone. or 9 ft.

Find the slant height of the conical top.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant

Find the slant height of the conical bottom. height of the cylinder. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

51.

Now find the area of the hexagon. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. The equilateral triangle can be divided into three

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

So, the area of the equilateral triangle is about 67.3 square feet. Use a trigonometric ratio to find the value of b. Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 Now use the area of the isosceles triangles to find 26.6 ft . the area of the equilateral triangle.

Therefore, the area of the shaded region is about 2 26.6 ft .

Use trigonometric ratios to find the values of b and h.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle) Use trigonometric ratios to find the values of b and h.

Therefore, the area of the shaded region is about 168.2 mm2.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid.

2. SOLUTION: 1. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a regular pentagon with area of the base and h is the height of the pyramid. sides of 4.4 centimeters and an apothem of 3 The base of this pyramid is a right triangle with legs centimeters. The height of the pyramid is 12 of 9 inches and 5 inches and the height of the centimeters. pyramid is 10 inches.

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 2. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of SOLUTION: the pyramid is 5.2 meters.

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

4. a square pyramid with a height of 14 meters and a

base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length Find the volume of each cone. Round to the of 8 meters and a width of 4.5 meters. The height of nearest tenth. the pyramid is 5.2 meters.

5. SOLUTION:

The volume of a circular cone is , or 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. The volume of a pyramid is , where B is the Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches. area of the base and h is the height of the pyramid.

The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

Find the volume of each cone. Round to the nearest tenth. 6. SOLUTION:

5. SOLUTION:

The volume of a circular cone is , or Use trigonometry to find the radius r.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius The volume of a circular cone is , or is or 3.5 inches. The height of the cone is 4 inches. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters 6. SOLUTION: SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the Use trigonometry to find the radius r. height is 10.5 millimeters.

The volume of a circular cone is , or

, where B is the area of the base, h is the

height of the cone, and r is the radius of the base. 8. a cone with a slant height of 25 meters and a radius The height of the cone is 11.5 centimeters. of 15 meters

SOLUTION:

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters Use the Pythagorean Theorem to find the height h of

SOLUTION: the cone. Then find its volume.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. So, the height of the cone is 20 meters.

8. a cone with a slant height of 25 meters and a radius of 15 meters 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical SOLUTION: building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

Use the Pythagorean Theorem to find the height h of The volume of a circular cone is , where B the cone. Then find its volume. is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

So, the height of the cone is 20 meters.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of 10. air that the heating and cooling systems would have SOLUTION: to accommodate. Round to the nearest tenth. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

CCSS SENSE-MAKING Find the volume of 11. each pyramid. Round to the nearest tenth if SOLUTION: necessary. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 12. 120°. The radius bisects the angle, so the right triangl SOLUTION: 90° triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 14. a pentagonal pyramid with a base area of 590 square 90° triangle. feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters The apothem is . and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and The volume of a pyramid is , where B is the hypotenuse 10.2 centimeters area of the base and h is the height of the pyramid. SOLUTION: Find the height of the right triangle.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

The length of the other leg of the right triangle is 6 cm.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a So, the area of the base B is 24 cm2. of the right triangle and then find the area of the

triangle. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

Therefore, the height of the triangular pyramid is 18 cm.

Find the volume of each cone. Round to the The length of the other leg of the right triangle is 6 nearest tenth. cm.

17. SOLUTION:

The volume of a circular cone is , 2 So, the area of the base B is 24 cm . where r is the radius of the base and h is the height of the cone. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Therefore, the height of the triangular pyramid is 18 cm. Therefore, the volume of the cone is about 235.6 in3.

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

18. 17. SOLUTION: SOLUTION: The volume of a circular cone is , where The volume of a circular cone is , r is the radius of the base and h is the height of the where r is the radius of the base and h is the height cone. The radius of this cone is 4.2 centimeters and of the cone. the height is 7.3 centimeters.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Therefore, the volume of the cone is about 134.8 3 cm .

Therefore, the volume of the cone is about 235.6 in3.

19. SOLUTION:

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where

Therefore, the volume of the cone is about 134.8 r is the radius of the base and h is the height of the 3 cone. The radius of this cone is 8 centimeters. cm .

19.

SOLUTION: Therefore, the volume of the cone is about 1473.1 cm3.

20.

Use a trigonometric ratio to find the height h of the SOLUTION: cone.

The volume of a circular cone is , where Use trigonometric ratios to find the height h and the r is the radius of the base and h is the height of the radius r of the cone. cone. The radius of this cone is 8 centimeters.

The volume of a circular cone is , where

Therefore, the volume of the cone is about 1473.1 r is the radius of the base and h is the height of the cone. cm3.

20. SOLUTION:

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches

SOLUTION: Use trigonometric ratios to find the height h and the radius r of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

The volume of a circular cone is , where r is the radius of the base and h is the height of the

cone. Therefore, the volume of the cone is about 1072.3

in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION:

The cone has a radius r of 1 centimeter and a slant

height of 5.6 centimeters. Use the Pythagorean 3 Therefore, the volume of the cone is about 2.8 ft . Theorem to find the height h of the cone. 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter 3 SOLUTION: Therefore, the volume of the cone is about 5.8 cm . The cone has a radius r of 1 centimeter and a slant 23. SNACKS Approximately how many cubic height of 5.6 centimeters. Use the Pythagorean centimeters of roasted peanuts will completely fill a Theorem to find the height h of the cone. paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The

height of the cone is 14 centimeters.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in 3 Therefore, the volume of the cone is about 5.8 cm . Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its 23. SNACKS Approximately how many cubic square base is 600 feet wide. Find the volume of this centimeters of roasted peanuts will completely fill a pyramid. paper cone that is 14 centimeters high and has a base SOLUTION: diameter of 8 centimeters? Round to the nearest

tenth. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its SOLUTION: square base is 600 feet wide. Find the volume of this pyramid. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with The volume of a pyramid is , where B is the sides of 2 feet. A central angle of the octagon is area of the base and h is the height of the pyramid. or 45°, so the angle formed in the triangle below is 22.5°.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the Use a trigonometric ratio to find the apothem a. greenhouse?

The height of this pyramid is 5 feet.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°. Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

Use a trigonometric ratio to find the apothem a. 26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

The height of this pyramid is 5 feet.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

Find the volume of each solid. Round to the nearest tenth. 27. SOLUTION:

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

28. SOLUTION:

27. SOLUTION: 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: 28. The building can be broken down into the rectangular SOLUTION: base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to The total volume is therefore 5000 + 1666.67 = purchase? 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base The volume of a pyramid is , where B is the is area of the base and h is the height of the pyramid.

The volume of the ceiling is It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters.

Describe how each change affects the volume of the

The total volume is therefore 5000 + 1666.67 = cone. a. The height is doubled. 6666.67 ft3. Two BTU's are needed for every cubic b. The radius is doubled. foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the 30. SCIENCE Refer to page 825. Determine the values. volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

a. Double h.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius The volume is doubled. of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the b. Double r. cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the

values. 2 The volume is multiplied by 2 or 4.

c. Double r and h.

a. Double h.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find The volume is doubled. the side length of the base. SOLUTION: b. Double r. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

12-5 Volumes of Pyramids and Cones The side length of the base is 15 cm. volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth 33. The volume of a cone is 196π cubic inches and the if necessary. height is 12 inches. What is the diameter? 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find SOLUTION: the side length of the base. The volume of a circular cone is , or SOLUTION: , where B is the area of the base, h is the The volume of a pyramid is , where B is the height of the cone, and r is the radius of the base. area of the base and h is the height of the pyramid. Since the diameter is 8 centimeters, the radius is 4 Let s be the side length of the base. centimeters.

The diameter is 2(7) or 14 inches. The side length of the base is 15 cm. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the 33. The volume of a cone is 196π cubic inches and the volume of the cone? height is 12 inches. What is the diameter? SOLUTION: SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

So, the radius is about 3.8 millimeters. The diameter is 2(7) or 14 inches. eSolutions Manual - Powered by Cognero Page 12 Use the Pythagorean Theorem to find the height of 34. The lateral area of a cone is 71.6 square millimeters the cone. and the slant height is 6 millimeters. What is the

volume of the cone? SOLUTION:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where The lateral area of a cone is , where r is r is the radius of the base and h is the height of the the radius and is the slant height of the cone. cone. Replace L with 71.6 and with 6, then solve for the

radius r.

Therefore, the volume of the cone is about 70.2 So, the radius is about 3.8 millimeters. 3 mm .

Use the Pythagorean Theorem to find the height of 35. MULTIPLE REPRESENTATIONS In this the cone. problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. So, the height of the cone is about 4.64 millimeters. SOLUTION: a. Use rectangular bases and pick values that The volume of a circular cone is , where multiply to make 24. r is the radius of the base and h is the height of the cone. Sample answer:

Therefore, the volume of the cone is about 70.2 3 mm .

c. ANALYTICAL Explain how multiplying the base b. The volumes are the same. The volume of a area and/or the height of the pyramid by 5 affects the pyramid equals one third times the base area times

volume of the pyramid. the height. So, if the base areas of two pyramids are SOLUTION: equal and their heights are equal, then their volumes are equal. a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

The volumes will only be equal when the radius of

the cone is equal to or when . 36. CCSS ARGUMENTS Determine whether the Therefore, the statement is true sometimes if the following statement is sometimes, always, or never base area of the cone is 3 times as great as the base true. Justify your reasoning. area of the prism. For example, if the base of the The volume of a cone with radius r and height h prism has an area of 10 square units, then its volume equals the volume of a prism with height h. is 10h cubic units. So, the cone must have a base SOLUTION: area of 30 square units so that its volume is

The volume of a cone with a radius r and height h is or 10h cubic units. . The volume of a prism with a height of h is where B is the area of the base of the 37. ERROR ANALYSIS Alexandra and Cornelio are prism. Set the volumes equal. calculating the volume of the cone below. Is either of them correct? Explain your answer.

SOLUTION: The slant height is used for surface area, but the The volumes will only be equal when the radius of height is used for volume. For this cone, the slant the cone is equal to or when . height of 13 is provided, and we need to calculate the height before we can calculate the volume. Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the Alexandra incorrectly used the slant height. prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base 38. REASONING A cone has a volume of 568 cubic area of 30 square units so that its volume is centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain or 10h cubic units. your reasoning. SOLUTION: 37. ERROR ANALYSIS Alexandra and Cornelio are 3 calculating the volume of the cone below. Is either of 1704 cm ; The formula for the volume of a cylinder them correct? Explain your answer. is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

OPEN ENDED 39. Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of SOLUTION: that. So, if a pyramid and prism have the same base, The slant height is used for surface area, but the then in order to have the same volume, the height of height is used for volume. For this cone, the slant the pyramid must be 3 times as great as the height of height of 13 is provided, and we need to calculate the the prism. height before we can calculate the volume.

Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain Set the base areas of the prism and pyramid, and your reasoning. make the height of the pyramid equal to 3 times the SOLUTION: height of the prism. 3 1704 cm ; The formula for the volume of a cylinder Sample answer: is V= Bh, while the formula for the volume of a cone A square pyramid with a base area of 16 and a is V = Bh. The volume of a cylinder is three times height of 12, a prism with a square base area of 16 as much as the volume of a cone with the same and a height of 4. radius and height. 40. WRITING IN MATH Compare and contrast 39. OPEN ENDED Give an example of a pyramid and finding volumes of pyramids and cones with finding a prism that have the same base and the same volumes of prisms and cylinders. volume. Explain your reasoning. SOLUTION: SOLUTION: To find the volume of each solid, you must know the The formula for volume of a prism is V = Bh and the area of the base and the height. The volume of a formula for the volume of a pyramid is one-third of pyramid is one third the volume of a prism that has that. So, if a pyramid and prism have the same base, the same height and base area. The volume of a then in order to have the same volume, the height of cone is one third the volume of a cylinder that has the the pyramid must be 3 times as great as the height of same height and base area. the prism. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

Set the base areas of the prism and pyramid, and A 12π make the height of the pyramid equal to 3 times the B 15π height of the prism.

C Sample answer: A square pyramid with a base area of 16 and a D height of 12, a prism with a square base area of 16 and a height of 4. SOLUTION: 40. WRITING IN MATH Compare and contrast Use the Pythagorean Theorem to find the radius r of finding volumes of pyramids and cones with finding the cone. volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

So, the radius of the cone is 3 centimeters.

A 12π The volume of a circular cone is , where B 15π r is the radius of the base and h is the height of the

C cone.

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

Therefore, the correct choice is A.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? F

SOLUTION: G The volume of a pyramid is , where B is the H area of the base and h is the height of the pyramid.

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the correct choice is F.

44. SAT/ACT For all

F A –8 B x – 4

G C D H E

J SOLUTION: SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5 So, the correct choice is E.

Find the volume of each prism.

45. SOLUTION:

So, the correct choice is F. The volume of a prism is , where B is the area of the base and h is the height of the prism. The 44. SAT/ACT For all base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the A –8 prism is 6 inches. B x – 4 C D E

SOLUTION: 3 Therefore, the volume of the prism is 1008 in .

So, the correct choice is E. 46. Find the volume of each prism. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base 45. of 10 feet and two legs of 13 feet. The height h will SOLUTION: bisect the base. Use the Pythagorean Theorem to find the height of the triangle. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

3 Therefore, the volume of the prism is 1008 in .

So, the height of the triangle is 12 feet. Find the area of the triangle.

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The 2 base of this prism is an isosceles triangle with a base So, the area of the base B is 60 ft . of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to The height h of the prism is 19 feet. find the height of the triangle.

3 Therefore, the volume of the prism is 1140 ft .

47.

So, the height of the triangle is 12 feet. Find the area SOLUTION: of the triangle. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet. Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin

3 allows the grain to be emptied more easily. Use the Therefore, the volume of the prism is 1140 ft . dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: 47. To find the entire surface area of the bin, find the surface area of the conical top and bottom and find SOLUTION: the surface area of the cylinder and add them. The volume of a prism is , where B is the The formula for finding the surface area of a cone is area of the base and h is the height of the prism. The , where is r is the radius and l is the slant height base of this prism is a rectangle with a length of 79.4 of the cone. meters and a width of 52.5 meters. The height of the prism is 102.3 meters. Find the slant height of the conical top.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface Find the slant height of the conical bottom. area of the bin with a conical top and bottom. Write The height of the conical bottom is 28 – (5 + 12 + 2) the exact answer and the answer rounded to the or 9 ft. nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top. The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50 The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. A regular hexagon has 6 sides, so the measure of the SOLUTION: interior angle is . The apothem bisects the Area of the shaded region = Area of the rectangle – angle, so we use 30° when using trig to find the Area of the circle length of the apothem. Area of the rectangle = 10(5) = 50

50. SOLUTION:

Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the Now find the area of the hexagon. length of the apothem.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

Now find the area of the hexagon. So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Find the area of the circle with a radius of 3.6 feet. Use a trigonometric ratio to find the value of b.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. Now use the area of the isosceles triangles to find the area of the equilateral triangle.

The equilateral triangle can be divided into three So, the area of the equilateral triangle is about 67.3 square feet. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the Subtract to find the area of shaded region. height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Therefore, the area of the shaded region is about 2 26.6 ft .

Use a trigonometric ratio to find the value of b.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral Now use the area of the isosceles triangles to find triangle plus the area of the inscribed circle. Let b the area of the equilateral triangle. represent the length of each side of the equilateral

triangle and h represent the radius of the inscribed circle.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown Therefore, the area of the shaded region is about below. 2 26.6 ft .

52. Use trigonometric ratios to find the values of b and SOLUTION: h. The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region. Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The A(shaded region) = A(circumscribed circle) - A angle in the triangle formed by the height.h of the (equilateral triangle) + A(inscribed circle) isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Therefore, the area of the shaded region is about 168.2 mm2.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

Find the volume of each pyramid.

1. 3. a rectangular pyramid with a height of 5.2 meters SOLUTION: and a base 8 meters by 4.5 meters The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the area of the base and h is the height of the pyramid. pyramid is 10 inches. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 2. The base of this pyramid is a square with sides of 8 SOLUTION: meters. The height of the pyramid is 14 meters.

5. SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters The volume of a circular cone is , or and a base 8 meters by 4.5 meters , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. The volume of a pyramid is , where B is the Since the diameter of this cone is 7 inches, the radius area of the base and h is the height of the pyramid. is or 3.5 inches. The height of the cone is 4 inches. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: 6. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

Use trigonometry to find the radius r.

The volume of a circular cone is , or

Find the volume of each cone. Round to the , where B is the area of the base, h is the nearest tenth. height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

5. SOLUTION:

The volume of a circular cone is , or 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius The volume of a circular cone is , or is or 3.5 inches. The height of the cone is 4 inches. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

6. 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: SOLUTION:

Use trigonometry to find the radius r.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

So, the height of the cone is 20 meters.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters

SOLUTION: 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical The volume of a circular cone is , or building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of , where B is the area of the base, h is the air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the SOLUTION: height is 10.5 millimeters. The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

10. SOLUTION:

The volume of a pyramid is , where B is the

So, the height of the cone is 20 meters. area of the base and h is the height of the pyramid.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if 12. necessary. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

11. The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he SOLUTION: 120°. The radius bisects the angle, so the right triangl 90° triangle. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is . 12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid. 15. a triangular pyramid with a height of 4.8 centimeters The base is a hexagon, so we need to make a right tri and a right triangle base with a leg 5 centimeters and determine the apothem. The interior angles of the he hypotenuse 10.2 centimeters 120°. The radius bisects the angle, so the right triangl 90° triangle. SOLUTION: Find the height of the right triangle.

The apothem is .

The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square area of the base and h is the height of the pyramid. feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg 15. a triangular pyramid with a height of 4.8 centimeters of 8 centimeters and a hypotenuse of 10 centimeters. and a right triangle base with a leg 5 centimeters and Use the Pythagorean Theorem to find the other leg a hypotenuse 10.2 centimeters of the right triangle and then find the area of the triangle. SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

The volume of a pyramid is , where B is the So, the area of the base B is 24 cm2. area of the base and h is the height of the pyramid. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height.

SOLUTION: Therefore, the height of the triangular pyramid is 18 The base of the pyramid is a right triangle with a leg cm. of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a Find the volume of each cone. Round to the of the right triangle and then find the area of the nearest tenth. triangle.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6

cm.

Therefore, the volume of the cone is about 235.6 in3.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. 18.

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 cm.

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

Therefore, the volume of the cone is about 134.8 3 cm . 17. SOLUTION:

The volume of a circular cone is ,

where r is the radius of the base and h is the height 19. of the cone. SOLUTION: Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Use a trigonometric ratio to find the height h of the cone.

Therefore, the volume of the cone is about 235.6 in3.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. 18. SOLUTION:

Therefore, the volume of the cone is about 134.8 20. 3 cm . SOLUTION:

19. Use trigonometric ratios to find the height h and the SOLUTION: radius r of the cone.

Use a trigonometric ratio to find the height h of the The volume of a circular cone is , where

cone. r is the radius of the base and h is the height of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

Therefore, the volume of the cone is about 1473.1 The volume of a circular cone is , where 3 cm . r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

20. SOLUTION:

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters Use trigonometric ratios to find the height h and the and a radius of 1 centimeter radius r of the cone. SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean

Theorem to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft . 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where 3 Therefore, the volume of the cone is about 5.8 cm . r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, 23. SNACKS Approximately how many cubic the radius is or 8 inches. centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 1072.3 cone. Since the diameter of the cone is 8 in 3. centimeters, the radius is or 4 centimeters. The 22. a right cone with a slant height of 5.6 centimeters height of the cone is 14 centimeters. and a radius of 1 centimeter

SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

3 Therefore, the volume of the cone is about 5.8 cm . 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base 23. SNACKS Approximately how many cubic has side lengths of 2 feet. What is the volume of the centimeters of roasted peanuts will completely fill a greenhouse? paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 SOLUTION: centimeters, the radius is or 4 centimeters. The The volume of a pyramid is , where B is the height of the cone is 14 centimeters. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with

sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its

square base is 600 feet wide. Find the volume of this Use a trigonometric ratio to find the apothem a. pyramid. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The height of this pyramid is 5 feet.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse? Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

SOLUTION: 26. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. Volume of the solid given = Volume of the small The base of the pyramid is a regular octagon with cone + Volume of the large cone sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet. 27.

SOLUTION:

Therefore, the volume of the greenhouse is about 3 32.2 ft .

Find the volume of each solid. Round to the nearest tenth. 28. SOLUTION:

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

SOLUTION: 27. The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base SOLUTION: is

The volume of the ceiling is

28. The total volume is therefore 5000 + 1666.67 = 3 SOLUTION: 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she area of the base and h is the height of the pyramid. needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the SOLUTION: cone. The building can be broken down into the rectangular a. The height is doubled. base and the pyramid ceiling. The volume of the base b. The radius is doubled. is c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

The volume of the ceiling is

a. Double h. The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the The volume is doubled. volume is helpful in this situation. b. SOLUTION: Double r.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2 The volume is multiplied by 2 or 4.

c. Double r and h. It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. volume is multiplied by 23 or 8. b. The radius is doubled. c. Both the radius and the height are doubled. Find each measure. Round to the nearest tenth if necessary. SOLUTION: 32. A square pyramid has a volume of 862.5 cubic Find the volume of the original cone. Then alter the centimeters and a height of 11.5 centimeters. Find values. the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled.

b. Double r.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

2 The volume is multiplied by 2 or 4. The volume of a circular cone is , or

c. Double r and h. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base.

SOLUTION: The volume of a pyramid is , where B is the The diameter is 2(7) or 14 inches. area of the base and h is the height of the pyramid. 34. The lateral area of a cone is 71.6 square millimeters Let s be the side length of the base. and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. The side length of the base is 15 cm. Replace L with 71.6 and with 6, then solve for the radius r.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

So, the radius is about 3.8 millimeters. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. Use the Pythagorean Theorem to find the height of Since the diameter is 8 centimeters, the radius is 4 the cone. centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the The diameter is 2(7) or 14 inches. cone.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters The lateral area of a cone is , where r is and a base area of 24 square centimeters. the radius and is the slant height of the cone. b. VERBAL What is true about the volumes of the Replace L with 71.6 and with 6, then solve for the two pyramids that you drew? Explain. radius r. c. ANALYTICAL Explain how multiplying the base

area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24. 12-5 Volumes of Pyramids and Cones

So, the radius is about 3.8 millimeters. Sample answer:

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

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the SOLUTION: volume is multiplied by 5. If both the base area and a. Use rectangular bases and pick values that the height are multiplied by 5, the volume is multiplied multiply to make 24. by 5 · 5 or 25.

Sample answer:

eSolutions Manual - Powered by Cognero Page 13

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of c. If the base area is multiplied by 5, the volume is h is where B is the area of the base of the multiplied by 5. If the height is multiplied by 5, the prism. Set the volumes equal. volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: SOLUTION: The volume of a cone with a radius r and height h is The slant height is used for surface area, but the . The volume of a prism with a height of height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the h is where B is the area of the base of the height before we can calculate the volume. prism. Set the volumes equal.

Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same The volumes will only be equal when the radius of radius and height. the cone is equal to or when . 39. OPEN ENDED Give an example of a pyramid and Therefore, the statement is true sometimes if the a prism that have the same base and the same base area of the cone is 3 times as great as the base volume. Explain your reasoning. area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume SOLUTION: is 10h cubic units. So, the cone must have a base The formula for volume of a prism is V = Bh and the area of 30 square units so that its volume is formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, or 10h cubic units. then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of 37. ERROR ANALYSIS Alexandra and Cornelio are the prism. calculating the volume of the cone below. Is either of them correct? Explain your answer.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a SOLUTION: height of 12, a prism with a square base area of 16 The slant height is used for surface area, but the and a height of 4. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the 40. WRITING IN MATH Compare and contrast height before we can calculate the volume. finding volumes of pyramids and cones with finding volumes of prisms and cylinders.

Alexandra incorrectly used the slant height. SOLUTION: To find the volume of each solid, you must know the 38. REASONING A cone has a volume of 568 cubic area of the base and the height. The volume of a centimeters. What is the volume of a cylinder that pyramid is one third the volume of a prism that has has the same radius and height as the cone? Explain the same height and base area. The volume of a your reasoning. cone is one third the volume of a cylinder that has the same height and base area. SOLUTION: 1704 cm3; The formula for the volume of a cylinder 41. A conical sand toy has the dimensions as shown is V= Bh, while the formula for the volume of a cone below. How many cubic centimeters of sand will it hold when it is filled to the top? is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same A 12π volume. Explain your reasoning. B 15π

SOLUTION: C The formula for volume of a prism is V = Bh and the

formula for the volume of a pyramid is one-third of D that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of SOLUTION: the pyramid must be 3 times as great as the height of the prism. Use the Pythagorean Theorem to find the radius r of the cone.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast So, the radius of the cone is 3 centimeters. finding volumes of pyramids and cones with finding volumes of prisms and cylinders. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the To find the volume of each solid, you must know the cone. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top? Therefore, the correct choice is A.

C The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. D

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters. F The volume of a circular cone is , where G r is the radius of the base and h is the height of the cone. H

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10

green} Number of possible outcomes : 25 Therefore, the correct choice is A. Favorable outcomes: {5 orange}

42. SHORT RESPONSE Brooke is buying a tent that Number of favorable outcomes: 5 is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the correct choice is F.

44. SAT/ACT For all

A –8 B x – 4 C D E 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows SOLUTION: the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the correct choice is E.

Find the volume of each prism.

G 45. H SOLUTION:

J The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 SOLUTION: inches and a width of 12 inches. The height h of the Possible outcomes: {6 red, 4 blue, 5 orange, 10 prism is 6 inches. green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

3 Therefore, the volume of the prism is 1008 in .

So, the correct choice is F.

44. SAT/ACT For all 46. SOLUTION: A –8 B x – 4 The volume of a prism is , where B is the area of the base and h is the height of the prism. The C base of this prism is an isosceles triangle with a base D of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to E find the height of the triangle.

SOLUTION:

So, the correct choice is E.

Find the volume of each prism.

45. So, the height of the triangle is 12 feet. Find the area of the triangle. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1008 in .

3 Therefore, the volume of the prism is 1140 ft .

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The 47. base of this prism is an isosceles triangle with a base SOLUTION: of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to The volume of a prism is , where B is the find the height of the triangle. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain So, the height of the triangle is 12 feet. Find the area after harvest. The cone at the bottom of the bin of the triangle. allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the So, the area of the base B is 60 ft2. surface area of the conical top and bottom and find the surface area of the cylinder and add them. The height h of the prism is 19 feet. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

3 Therefore, the volume of the prism is 1140 ft .

47. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The Find the slant height of the conical bottom. base of this prism is a rectangle with a length of 79.4 The height of the conical bottom is 28 – (5 + 12 + 2) meters and a width of 52.5 meters. The height of the or 9 ft. prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin The formula for finding the surface area of a cylinder allows the grain to be emptied more easily. Use the is , where is r is the radius and h is the slant dimensions in the diagram to find the entire surface height of the cylinder. area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. Surface area of the bin = Surface area of the SOLUTION: cylinder + Surface area of the conical top + surface To find the entire surface area of the bin, find the area of the conical bottom. surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top. Find the area of each shaded region. Polygons

in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

50. SOLUTION:

Area of the shaded region = Area of the circle – Area of the hexagon

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the Surface area of the bin = Surface area of the angle, so we use 30° when using trig to find the cylinder + Surface area of the conical top + surface length of the apothem. area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the 51. angle, so we use 30° when using trig to find the length of the apothem. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. So, the area of the equilateral triangle is about 67.3 Find the area of the circle with a radius of 3.6 feet. square feet.

Subtract to find the area of shaded region.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. Therefore, the area of the shaded region is about 2 26.6 ft .

52. SOLUTION: The equilateral triangle can be divided into three The area of the shaded region is the area of the isosceles triangles with central angles of or circumscribed circle minus the area of the equilateral 120°, so the angle in the triangle created by the triangle plus the area of the inscribed circle. Let b height of the isosceles triangle is 60° and the base is represent the length of each side of the equilateral half the length b of the side of the equilateral triangle. triangle and h represent the radius of the inscribed circle.

Use a trigonometric ratio to find the value of b.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below. Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and So, the area of the equilateral triangle is about 67.3 h. square feet.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 26.6 ft . The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A 52. (equilateral triangle) + A(inscribed circle) SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed Therefore, the area of the shaded region is about circle. 168.2 mm2.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid. 2. SOLUTION:

The volume of a pyramid is , where B is the 1. area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with SOLUTION: sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 The volume of a pyramid is , where B is the centimeters. area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the Find the volume of each cone. Round to the nearest tenth. area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the 4. a square pyramid with a height of 14 meters and a height of the cone, and r is the radius of the base. base with 8-meter side lengths Since the diameter of this cone is 7 inches, the radius SOLUTION: is or 3.5 inches. The height of the cone is 4 inches.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

Find the volume of each cone. Round to the 6. nearest tenth. SOLUTION:

5. SOLUTION: Use trigonometry to find the radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the The volume of a circular cone is , or height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius , where B is the area of the base, h is the is or 3.5 inches. The height of the cone is 4 inches. height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or 6. SOLUTION: , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

Use trigonometry to find the radius r.

The volume of a circular cone is , or 8. a cone with a slant height of 25 meters and a radius , where B is the area of the base, h is the of 15 meters height of the cone, and r is the radius of the base. SOLUTION: The height of the cone is 11.5 centimeters.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. 7. an oblique cone with a height of 10.5 millimeters and

a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. So, the height of the cone is 20 meters. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical 8. a cone with a slant height of 25 meters and a radius building. If the height is 100 feet and the area of the of 15 meters base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have SOLUTION: to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. Use the Pythagorean Theorem to find the height h of For this cone, the area of the base is 15,400 square

the cone. Then find its volume. feet and the height is 100 feet.

So, the height of the cone is 20 meters. CCSS SENSE-MAKING Find the volume of

each pyramid. Round to the nearest tenth if necessary.

9. MUSEUMS The sky dome of the National Corvette 10. Museum in Bowling Green, Kentucky, is a conical SOLUTION: building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of The volume of a pyramid is , where B is the air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. area of the base and h is the height of the pyramid.

SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

11. SOLUTION: CCSS SENSE-MAKING Find the volume of The volume of a pyramid is , where B is the each pyramid. Round to the nearest tenth if necessary. area of the base and h is the height of the pyramid.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle. 12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid. 14. a pentagonal pyramid with a base area of 590 square The base is a hexagon, so we need to make a right tri feet and an altitude of 7 feet determine the apothem. The interior angles of the he SOLUTION: 120°. The radius bisects the angle, so the right triangl 90° triangle. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: The apothem is . Find the height of the right triangle.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume of a pyramid is , where B is the

15. a triangular pyramid with a height of 4.8 centimeters area of the base and h is the height of the pyramid. and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The length of the other leg of the right triangle is 6 cm.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: So, the area of the base B is 24 cm2. The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a Replace V with 144 and B with 24 in the formula for of the right triangle and then find the area of the the volume of a pyramid and solve for the height h. triangle.

Therefore, the height of the triangular pyramid is 18 cm.

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

The length of the other leg of the right triangle is 6 cm. 17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

2 So, the area of the base B is 24 cm . Since the diameter of this cone is 10 inches, the

radius is or 5 inches. The height of the cone is 9 Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. inches.

Therefore, the volume of the cone is about 235.6 in3.

Therefore, the height of the triangular pyramid is 18 cm.

Find the volume of each cone. Round to the nearest tenth. 18. SOLUTION:

The volume of a circular cone is , where 17. r is the radius of the base and h is the height of the SOLUTION: cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches. Therefore, the volume of the cone is about 134.8 3 cm .

Therefore, the volume of the cone is about 235.6 in3. 19. SOLUTION:

18. SOLUTION:

The volume of a circular cone is , where Use a trigonometric ratio to find the height h of the r is the radius of the base and h is the height of the cone. cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Therefore, the volume of the cone is about 134.8 3 cm .

Therefore, the volume of the cone is about 1473.1 19. cm3. SOLUTION:

20. SOLUTION:

Use a trigonometric ratio to find the height h of the cone.

Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 1473.1 cm3.

20. 3 Therefore, the volume of the cone is about 2.8 ft . SOLUTION: 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the Use trigonometric ratios to find the height h and the cone. Since the diameter of this cone is 16 inches, radius r of the cone. the radius is or 8 inches.

The volume of a circular cone is , where Therefore, the volume of the cone is about 1072.3 r is the radius of the base and h is the height of the 3 cone. in .

22. a right cone with a slant height of 5.6 centimeters

and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 in 3. 3 Therefore, the volume of the cone is about 5.8 cm . 22. a right cone with a slant height of 5.6 centimeters

and a radius of 1 centimeter 23. SNACKS Approximately how many cubic SOLUTION: centimeters of roasted peanuts will completely fill a The cone has a radius r of 1 centimeter and a slant paper cone that is 14 centimeters high and has a base height of 5.6 centimeters. Use the Pythagorean diameter of 8 centimeters? Round to the nearest Theorem to find the height h of the cone. tenth. SOLUTION:

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

23. SNACKS Approximately how many cubic SOLUTION: centimeters of roasted peanuts will completely fill a The volume of a pyramid is , where B is the paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest area of the base and h is the height of the pyramid. tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The 25. GARDENING The greenhouse is a regular height of the cone is 14 centimeters. octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in SOLUTION: Memphis, Tennessee, is the third largest pyramid in The volume of a pyramid is , where B is the the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this area of the base and h is the height of the pyramid. pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is SOLUTION: or 45°, so the angle formed in the triangle The volume of a pyramid is , where B is the below is 22.5°.

area of the base and h is the height of the pyramid.

Use a trigonometric ratio to find the apothem a. 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

The height of this pyramid is 5 feet.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is Therefore, the volume of the greenhouse is about 3 or 45°, so the angle formed in the triangle 32.2 ft . below is 22.5°. Find the volume of each solid. Round to the nearest tenth.

26. SOLUTION: Use a trigonometric ratio to find the apothem a. Volume of the solid given = Volume of the small cone + Volume of the large cone

The height of this pyramid is 5 feet.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

Find the volume of each solid. Round to the 27. nearest tenth. SOLUTION:

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

28. SOLUTION:

29. HEATING Sam is building an art studio in her 27. backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) SOLUTION: required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

28. SOLUTION:

The volume of the ceiling is

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) The total volume is therefore 5000 + 1666.67 = required to heat the building. For new construction 6666.67 ft3. Two BTU's are needed for every cubic with good insulation, there should be 2 BTUs per foot, so the size of the heating unit Sam should buy is cubic foot. What size unit does Sam need to 6666.67 × 2 = 13,333 BTUs. purchase?

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the SOLUTION:

The building can be broken down into the rectangular area of the base and h is the height of the pyramid. base and the pyramid ceiling. The volume of the base is

It tells Marta how much clay is needed to make the The volume of the ceiling is model. 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. The total volume is therefore 5000 + 1666.67 = c. Both the radius and the height are doubled. 3 6666.67 ft . Two BTU's are needed for every cubic SOLUTION: foot, so the size of the heating unit Sam should buy is Find the volume of the original cone. Then alter the 6666.67 × 2 = 13,333 BTUs. values.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. a. Double h.

It tells Marta how much clay is needed to make the The volume is doubled. model. b. Double r. 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled.

2 SOLUTION: The volume is multiplied by 2 or 4. Find the volume of the original cone. Then alter the values. c. Double r and h.

volume is multiplied by 23 or 8. a. Double h. Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume is doubled. The volume of a pyramid is , where B is the

b. Double r. area of the base and h is the height of the pyramid. Let s be the side length of the base.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the volume is multiplied by 23 or 8. height is 12 inches. What is the diameter?

Find each measure. Round to the nearest tenth SOLUTION: if necessary. 32. A square pyramid has a volume of 862.5 cubic The volume of a circular cone is , or centimeters and a height of 11.5 centimeters. Find the side length of the base. , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 The volume of a pyramid is , where B is the centimeters.

area of the base and h is the height of the pyramid. Let s be the side length of the base.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the The side length of the base is 15 cm. volume of the cone? SOLUTION: 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the

height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 The lateral area of a cone is , where r is centimeters. the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of

the cone.

The diameter is 2(7) or 14 inches. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

Therefore, the volume of the cone is about 70.2 3 mm .

MULTIPLE REPRESENTATIONS So, the radius is about 3.8 millimeters. 35. In this

problem, you will investigate rectangular pyramids. a. GEOMETRIC Use the Pythagorean Theorem to find the height of Draw two pyramids with the cone. different bases that have a height of 10 centimeters

and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that So, the height of the cone is about 4.64 millimeters. multiply to make 24.

Sample answer: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the b. The volumes are the same. The volume of a two pyramids that you drew? Explain. pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are c. ANALYTICAL Explain how multiplying the base equal and their heights are equal, then their volumes area and/or the height of the pyramid by 5 affects the are equal. volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the 12-5volume Volumes is multiplied of Pyramids by 5. and If both Cones the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of

h is where B is the area of the base of the prism. Set the volumes equal.

The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the CCSS ARGUMENTS 36. Determine whether the prism has an area of 10 square units, then its volume following statement is sometimes, always, or never is 10h cubic units. So, the cone must have a base

true. Justify your reasoning. area of 30 square units so that its volume is The volume of a cone with radius r and height h equals the volume of a prism with height h. or 10h cubic units. SOLUTION: The volume of a cone with a radius r and height h is 37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of . The volume of a prism with a height of them correct? Explain your answer. h is where B is the area of the base of the prism. Set the volumes equal.

SOLUTION: The slant height is used for surface area, but the eSolutions Manual - Powered by Cognero height is used for volume. For this cone, the slantPage 14 height of 13 is provided, and we need to calculate the height before we can calculate the volume. The volumes will only be equal when the radius of the cone is equal to or when . Alexandra incorrectly used the slant height. Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base 38. REASONING A cone has a volume of 568 cubic area of the prism. For example, if the base of the centimeters. What is the volume of a cylinder that prism has an area of 10 square units, then its volume has the same radius and height as the cone? Explain is 10h cubic units. So, the cone must have a base your reasoning. area of 30 square units so that its volume is SOLUTION: or 10h cubic units. 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone 37. ERROR ANALYSIS Alexandra and Cornelio are is V = Bh. The volume of a cylinder is three times calculating the volume of the cone below. Is either of as much as the volume of a cone with the same them correct? Explain your answer. radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of SOLUTION: the pyramid must be 3 times as great as the height of the prism. The slant height is used for surface area, but the

height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic Set the base areas of the prism and pyramid, and centimeters. What is the volume of a cylinder that make the height of the pyramid equal to 3 times the

has the same radius and height as the cone? Explain height of the prism.

your reasoning. Sample answer: SOLUTION: A square pyramid with a base area of 16 and a 3 height of 12, a prism with a square base area of 16 1704 cm ; The formula for the volume of a cylinder and a height of 4. is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times 40. WRITING IN MATH Compare and contrast as much as the volume of a cone with the same finding volumes of pyramids and cones with finding radius and height. volumes of prisms and cylinders.

39. OPEN ENDED Give an example of a pyramid and SOLUTION: a prism that have the same base and the same To find the volume of each solid, you must know the volume. Explain your reasoning. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has SOLUTION: the same height and base area. The volume of a The formula for volume of a prism is V = Bh and the cone is one third the volume of a cylinder that has the formula for the volume of a pyramid is one-third of same height and base area. that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of 41. A conical sand toy has the dimensions as shown the pyramid must be 3 times as great as the height of below. How many cubic centimeters of sand will it the prism. hold when it is filled to the top?

A 12π B 15π

Set the base areas of the prism and pyramid, and C make the height of the pyramid equal to 3 times the height of the prism. D Sample answer: A square pyramid with a base area of 16 and a SOLUTION: height of 12, a prism with a square base area of 16 Use the Pythagorean Theorem to find the radius r of and a height of 4. the cone.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top? So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where

r is the radius of the base and h is the height of the A 12π cone. B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of Therefore, the correct choice is A. the cone. 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. PROBABILITY 43. A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that F is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, G how tall is the tent’s center pole?

SOLUTION: H

The volume of a pyramid is , where B is the J area of the base and h is the height of the pyramid. SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

44. SAT/ACT For all

A –8 B x – 4 C F D

G E

H SOLUTION:

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} So, the correct choice is E. Number of possible outcomes : 25 Favorable outcomes: {5 orange} Find the volume of each prism. Number of favorable outcomes: 5

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 So, the correct choice is F. inches and a width of 12 inches. The height h of the prism is 6 inches. 44. SAT/ACT For all

A –8 B x – 4 C

D 3 E Therefore, the volume of the prism is 1008 in .

SOLUTION:

46.

So, the correct choice is E. SOLUTION: The volume of a prism is , where B is the Find the volume of each prism. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle. 45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the height of the triangle is 12 feet. Find the area 3 Therefore, the volume of the prism is 1008 in . of the triangle.

46.

SOLUTION: So, the area of the base B is 60 ft2. The volume of a prism is , where B is the area of the base and h is the height of the prism. The The height h of the prism is 19 feet. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

3 Therefore, the volume of the prism is 1140 ft .

47. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The So, the height of the triangle is 12 feet. Find the area base of this prism is a rectangle with a length of 79.4 of the triangle. meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

2 Therefore, the volume of the prism is about 426,437.6 So, the area of the base B is 60 ft . 3 m . The height h of the prism is 19 feet. 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write 3 the exact answer and the answer rounded to the Therefore, the volume of the prism is 1140 ft . nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. 47. The formula for finding the surface area of a cone is SOLUTION: , where is r is the radius and l is the slant height of the cone. The volume of a prism is , where B is the area of the base and h is the height of the prism. The Find the slant height of the conical top. base of this prism is a rectangle with a length of 79.4

meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain Find the slant height of the conical bottom. after harvest. The cone at the bottom of the bin The height of the conical bottom is 28 – (5 + 12 + 2) allows the grain to be emptied more easily. Use the or 9 ft. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. The formula for finding the surface area of a cylinder Find the slant height of the conical top. is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) Find the area of each shaded region. Polygons or 9 ft. in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle

Area of the rectangle = 10(5) = 50

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface 50. area of the conical bottom. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

Find the area of each shaded region. Polygons in 50 - 52 are regular. A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the 49. angle, so we use 30° when using trig to find the length of the apothem. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the Now find the area of the hexagon.

interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. Now find the area of the hexagon.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. Use a trigonometric ratio to find the value of b.

Find the area of the circle with a radius of 3.6 feet.

Now use the area of the isosceles triangles to find So, the area of the circle is about 40.7 square feet. the area of the equilateral triangle.

Next, find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet.

The equilateral triangle can be divided into three Subtract to find the area of shaded region.

isosceles triangles with central angles of or

120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle. Therefore, the area of the shaded region is about 2 26.6 ft .

Use a trigonometric ratio to find the value of b. 52.

SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral Now use the area of the isosceles triangles to find triangle and h represent the radius of the inscribed the area of the equilateral triangle. circle.

So, the area of the equilateral triangle is about 67.3 Divide the equilateral triangle into three isosceles square feet. triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the Subtract to find the area of shaded region. isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Therefore, the area of the shaded region is about 2 26.6 ft .

Use trigonometric ratios to find the values of b and h. 52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle) Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below. Therefore, the area of the shaded region is about

168.2 mm2.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

The volume of a pyramid is , where B is the Find the volume of each pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters. 1.

SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

2. SOLUTION:

The volume of a pyramid is , where B is the 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with SOLUTION: sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 The volume of a pyramid is , where B is the centimeters. area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

3. a rectangular pyramid with a height of 5.2 meters Find the volume of each cone. Round to the and a base 8 meters by 4.5 meters nearest tenth. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 5. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of SOLUTION: the pyramid is 5.2 meters. The volume of a circular cone is , or

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

6. SOLUTION:

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

Use trigonometry to find the radius r.

5. SOLUTION: The volume of a circular cone is , or

The volume of a circular cone is , or , where B is the area of the base, h is the

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. 6. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. SOLUTION:

Use trigonometry to find the radius r.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: So, the height of the cone is 20 meters. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. 8. a cone with a slant height of 25 meters and a radius SOLUTION: of 15 meters The volume of a circular cone is , where B SOLUTION: is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

So, the height of the cone is 20 meters.

10. SOLUTION:

The volume of a pyramid is , where B is the 9. MUSEUMS The sky dome of the National Corvette area of the base and h is the height of the pyramid. Museum in Bowling Green, Kentucky, is a conical

building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

11.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION: 12. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

11. SOLUTION: 13. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

13. SOLUTION: 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet The volume of a pyramid is , where B is the SOLUTION: base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base is a hexagon, so we need to make a right tri area of the base and h is the height of the pyramid. determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl

90° triangle.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: 16. A triangular pyramid with a right triangle base with a Find the height of the right triangle. leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

The volume of a pyramid is , where B is the The length of the other leg of the right triangle is 6 area of the base and h is the height of the pyramid. cm.

2 16. A triangular pyramid with a right triangle base with a So, the area of the base B is 24 cm . leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

Therefore, the height of the triangular pyramid is 18 cm.

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

17. SOLUTION: The length of the other leg of the right triangle is 6 cm. The volume of a circular cone is ,

where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9

inches.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

Therefore, the volume of the cone is about 235.6 in3.

Therefore, the height of the triangular pyramid is 18 18. cm. SOLUTION: Find the volume of each cone. Round to the nearest tenth. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

17.

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Therefore, the volume of the cone is about 134.8 3 Since the diameter of this cone is 10 inches, the cm . radius is or 5 inches. The height of the cone is 9 inches.

19. SOLUTION:

Therefore, the volume of the cone is about 235.6 in3.

Use a trigonometric ratio to find the height h of the 18. cone. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Therefore, the volume of the cone is about 134.8 3 cm . Therefore, the volume of the cone is about 1473.1 cm3.

19. SOLUTION: 20. SOLUTION:

Use a trigonometric ratio to find the height h of the Use trigonometric ratios to find the height h and the cone. radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 1473.1 cm3.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches 20. SOLUTION: SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1072.3 in 3.

The volume of a circular cone is , where 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter r is the radius of the base and h is the height of the cone. SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 5.8 cm .

Therefore, the volume of the cone is about 1072.3 23. SNACKS Approximately how many cubic in 3. centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base 22. a right cone with a slant height of 5.6 centimeters diameter of 8 centimeters? Round to the nearest and a radius of 1 centimeter tenth. SOLUTION: SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean The volume of a circular cone is , where Theorem to find the height h of the cone. r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

The volume of a pyramid is , where B is the 3 Therefore, the volume of the cone is about 5.8 cm . area of the base and h is the height of the pyramid. 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where 25. GARDENING The greenhouse is a regular r is the radius of the base and h is the height of the octagonal pyramid with a height of 5 feet. The base cone. Since the diameter of the cone is 8 has side lengths of 2 feet. What is the volume of the greenhouse? centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

SOLUTION:

Therefore, the paper cone will hold about 234.6 cm3 The volume of a pyramid is , where B is the of roasted peanuts. area of the base and h is the height of the pyramid. 24. CCSS MODELING The Pyramid Arena in The base of the pyramid is a regular octagon with Memphis, Tennessee, is the third largest pyramid in sides of 2 feet. A central angle of the octagon is the world. It is approximately 350 feet tall, and its or 45°, so the angle formed in the triangle square base is 600 feet wide. Find the volume of this below is 22.5°. pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Use a trigonometric ratio to find the apothem a.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base The height of this pyramid is 5 feet. has side lengths of 2 feet. What is the volume of the greenhouse?

SOLUTION: Therefore, the volume of the greenhouse is about 3 The volume of a pyramid is , where B is the 32.2 ft . area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with Find the volume of each solid. Round to the sides of 2 feet. A central angle of the octagon is nearest tenth. or 45°, so the angle formed in the triangle below is 22.5°.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet.

27. Therefore, the volume of the greenhouse is about SOLUTION: 3 32.2 ft .

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

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

28. SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

28.

SOLUTION: The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 =

6666.67 ft3. Two BTU's are needed for every cubic 29. HEATING Sam is building an art studio in her foot, so the size of the heating unit Sam should buy is backyard. To buy a heating unit for the space, she 6666.67 × 2 = 13,333 BTUs. needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction SCIENCE with good insulation, there should be 2 BTUs per 30. Refer to page 825. Determine the cubic foot. What size unit does Sam need to volume of the model. Explain why knowing the

purchase? volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the The volume of the ceiling is cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values. The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the

volume of the model. Explain why knowing the volume is helpful in this situation. a. Double h. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume is doubled.

b. Double r.

It tells Marta how much clay is needed to make the model.

b. The radius is doubled. c. Double r and h. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic a. Double h. centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The volume is doubled.

b. Double r.

2 The volume is multiplied by 2 or 4.

c. Double r and h. The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or volume is multiplied by 23 or 8. , where B is the area of the base, h is the Find each measure. Round to the nearest tenth if necessary. height of the cone, and r is the radius of the base. 32. A square pyramid has a volume of 862.5 cubic Since the diameter is 8 centimeters, the radius is 4 centimeters and a height of 11.5 centimeters. Find centimeters. the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or The lateral area of a cone is , where r is the radius and is the slant height of the cone. , where B is the area of the base, h is the Replace L with 71.6 and with 6, then solve for the height of the cone, and r is the radius of the base. radius r. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the radius is about 3.8 millimeters.

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

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the

volume of the cone? So, the height of the cone is about 4.64 millimeters. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r. Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. So, the radius is about 3.8 millimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. Use the Pythagorean Theorem to find the height of c. ANALYTICAL Explain how multiplying the base the cone. area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

MULTIPLE REPRESENTATIONS 35. In this b. The volumes are the same. The volume of a

problem, you will investigate rectangular pyramids. pyramid equals one third times the base area times a. GEOMETRIC Draw two pyramids with the height. So, if the base areas of two pyramids are different bases that have a height of 10 centimeters equal and their heights are equal, then their volumes and a base area of 24 square centimeters. are equal. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

c. If the base area is multiplied by 5, the volume is CCSS ARGUMENTS multiplied by 5. If the height is multiplied by 5, the 36. Determine whether the volume is multiplied by 5. If both the base area and following statement is sometimes, always, or never

the height are multiplied by 5, the volume is multiplied true. Justify your reasoning. by 5 · 5 or 25. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

36. CCSS ARGUMENTS Determine whether the or 10h cubic units. following statement is sometimes, always, or never true. Justify your reasoning. 37. ERROR ANALYSIS Alexandra and Cornelio are The volume of a cone with radius r and height h calculating the volume of the cone below. Is either of equals the volume of a prism with height h. them correct? Explain your answer. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

Alexandra incorrectly used the slant height.

The volumes will only be equal when the radius of 38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that the cone is equal to or when . has the same radius and height as the cone? Explain Therefore, the statement is true sometimes if the your reasoning. base area of the cone is 3 times as great as the base SOLUTION: area of the prism. For example, if the base of the 3 prism has an area of 10 square units, then its volume 1704 cm ; The formula for the volume of a cylinder is 10h cubic units. So, the cone must have a base is V= Bh, while the formula for the volume of a cone area of 30 square units so that its volume is is V = Bh. The volume of a cylinder is three times

12-5 Volumes or 10 of Pyramidsh cubic units. and Cones as much as the volume of a cone with the same radius and height.

37. ERROR ANALYSIS Alexandra and Cornelio are 39. OPEN ENDED Give an example of a pyramid and calculating the volume of the cone below. Is either of a prism that have the same base and the same them correct? Explain your answer. volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume. Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. Alexandra incorrectly used the slant height. Sample answer: 38. REASONING A cone has a volume of 568 cubic A square pyramid with a base area of 16 and a centimeters. What is the volume of a cylinder that height of 12, a prism with a square base area of 16 has the same radius and height as the cone? Explain and a height of 4. your reasoning. 40. WRITING IN MATH Compare and contrast SOLUTION: finding volumes of pyramids and cones with finding 1704 cm3; The formula for the volume of a cylinder volumes of prisms and cylinders. is V= Bh, while the formula for the volume of a cone SOLUTION: is V = Bh. The volume of a cylinder is three times To find the volume of each solid, you must know the as much as the volume of a cone with the same area of the base and the height. The volume of a radius and height. pyramid is one third the volume of a prism that has the same height and base area. The volume of a 39. OPEN ENDED Give an example of a pyramid and cone is one third the volume of a cylinder that has the a prism that have the same base and the same same height and base area. volume. Explain your reasoning. SOLUTION: 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it The formula for volume of a prism is V = Bh and the hold when it is filled to the top? formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism. A 12π B 15π

D eSolutions Manual - Powered by Cognero Page 15 Set the base areas of the prism and pyramid, and SOLUTION: make the height of the pyramid equal to 3 times the Use the Pythagorean Theorem to find the radius r of height of the prism. the cone.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION:

same height and base area. So, the radius of the cone is 3 centimeters. 41. A conical sand toy has the dimensions as shown The volume of a circular cone is , where below. How many cubic centimeters of sand will it hold when it is filled to the top? r is the radius of the base and h is the height of the cone.

A 12π B 15π

C Therefore, the correct choice is A. D 42. SHORT RESPONSE Brooke is buying a tent that SOLUTION: is in the shape of a rectangular pyramid. The base is Use the Pythagorean Theorem to find the radius r of 6 feet by 8 feet. If the tent holds 88 cubic feet of air,

the cone. how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the radius of the cone is 3 centimeters. 43. PROBABILITY A spinner has sections colored The volume of a circular cone is , where red, blue, orange, and green. The table below shows the results of several spins. What is the experimental r is the radius of the base and h is the height of the probability of the spinner landing on orange? cone.

G Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that H is in the shape of a rectangular pyramid. The base is

6 feet by 8 feet. If the tent holds 88 cubic feet of air, J how tall is the tent’s center pole? SOLUTION: SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 The volume of a pyramid is , where B is the green} Number of possible outcomes : 25 area of the base and h is the height of the pyramid. Favorable outcomes: {5 orange} Number of favorable outcomes: 5

So, the correct choice is F.

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows 44. SAT/ACT For all the results of several spins. What is the experimental probability of the spinner landing on orange? A –8 B x – 4 C D E

F SOLUTION:

J So, the correct choice is E. Find the volume of each prism. SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} 45. Number of favorable outcomes: 5 SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is F.

44. SAT/ACT For all

A –8 3 B x – 4 Therefore, the volume of the prism is 1008 in . C D E

SOLUTION: 46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base So, the correct choice is E. of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to Find the volume of each prism. find the height of the triangle.

45.

So, the height of the triangle is 12 feet. Find the area of the triangle.

3 Therefore, the volume of the prism is 1008 in .

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet. 46.

SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will 3 bisect the base. Use the Pythagorean Theorem to Therefore, the volume of the prism is 1140 ft . find the height of the triangle.

So, the height of the triangle is 12 feet. Find the area of the triangle.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain 2 So, the area of the base B is 60 ft . after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the The height h of the prism is 19 feet. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847.

SOLUTION: 3 To find the entire surface area of the bin, find the Therefore, the volume of the prism is 1140 ft . surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

47. Find the slant height of the conical top.

Therefore, the volume of the prism is about 426,437.6 Find the slant height of the conical bottom. 3 The height of the conical bottom is 28 – (5 + 12 + 2) m . or 9 ft.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find The formula for finding the surface area of a cylinder the surface area of the cylinder and add them. is , where is r is the radius and h is the slant The formula for finding the surface area of a cone is height of the cylinder. , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top. Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2)

or 9 ft. 49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder. 50. SOLUTION:

Surface area of the bin = Surface area of the Area of the shaded region = Area of the circle –

cylinder + Surface area of the conical top + surface Area of the hexagon area of the conical bottom.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the Find the area of each shaded region. Polygons angle, so we use 30° when using trig to find the in 50 - 52 are regular. length of the apothem.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

Now find the area of the hexagon.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

51.

SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b. 51.

SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

The equilateral triangle can be divided into three Therefore, the area of the shaded region is about isosceles triangles with central angles of or 2 120°, so the angle in the triangle created by the 26.6 ft . height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

52. SOLUTION:

The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral Use a trigonometric ratio to find the value of b. triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the So, the area of the equilateral triangle is about 67.3 length of the side of the equilateral triangle as shown square feet. below.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about Use trigonometric ratios to find the values of b and 2 26.6 ft . h.

52.

SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral

triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral The area of the equilateral triangle will equal three triangle and h represent the radius of the inscribed times the area of any of the isosceles triangles. Find

circle. the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The Therefore, the area of the shaded region is about angle in the triangle formed by the height.h of the 168.2 mm2. isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

Find the volume of each pyramid.

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters 1. SOLUTION:

SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length area of the base and h is the height of the pyramid. of 8 meters and a width of 4.5 meters. The height of The base of this pyramid is a right triangle with legs the pyramid is 5.2 meters. of 9 inches and 5 inches and the height of the pyramid is 10 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

2. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 Find the volume of each cone. Round to the centimeters. The height of the pyramid is 12 nearest tenth. centimeters.

5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the 3. a rectangular pyramid with a height of 5.2 meters height of the cone, and r is the radius of the base. and a base 8 meters by 4.5 meters Since the diameter of this cone is 7 inches, the radius SOLUTION: is or 3.5 inches. The height of the cone is 4 inches.

6. 4. a square pyramid with a height of 14 meters and a SOLUTION: base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 Use trigonometry to find the radius r. meters. The height of the pyramid is 14 meters.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. Find the volume of each cone. Round to the nearest tenth.

5. 7. an oblique cone with a height of 10.5 millimeters and SOLUTION: a radius of 1.6 millimeters

The volume of a circular cone is , or SOLUTION: The volume of a circular cone is , or , where B is the area of the base, h is the height of the cone, and r is the radius of the base. , where B is the area of the base, h is the Since the diameter of this cone is 7 inches, the radius height of the cone, and r is the radius of the base. is or 3.5 inches. The height of the cone is 4 inches. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: 6. SOLUTION:

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

Use trigonometry to find the radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the So, the height of the cone is 20 meters. height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

9. MUSEUMS The sky dome of the National Corvette 7. an oblique cone with a height of 10.5 millimeters and Museum in Bowling Green, Kentucky, is a conical a radius of 1.6 millimeters building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of SOLUTION: air that the heating and cooling systems would have The volume of a circular cone is , or to accommodate. Round to the nearest tenth. SOLUTION: , where B is the area of the base, h is the The volume of a circular cone is , where B height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the is the area of the base and h is the height of the height is 10.5 millimeters. cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

8. a cone with a slant height of 25 meters and a radius of 15 meters CCSS SENSE-MAKING Find the volume of SOLUTION: each pyramid. Round to the nearest tenth if necessary.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. 10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the height of the cone is 20 meters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical 11. building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of SOLUTION: air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid.

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

12. SOLUTION: CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if The volume of a pyramid is , where B is the necessary. area of the base and h is the height of the pyramid.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle. 11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION: 15. a triangular pyramid with a height of 4.8 centimeters The volume of a pyramid is , where B is the and a right triangle base with a leg 5 centimeters and base and h is the height of the pyramid. hypotenuse 10.2 centimeters SOLUTION: The base is a hexagon, so we need to make a right tri Find the height of the right triangle. determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the 16. A triangular pyramid with a right triangle base with a area of the base and h is the height of the pyramid. leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height.

SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for The volume of a pyramid is , where B is the the volume of a pyramid and solve for the height h.

area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a Therefore, the height of the triangular pyramid is 18 leg 8 centimeters and hypotenuse 10 centimeters has cm. a volume of 144 cubic centimeters. Find the height. Find the volume of each cone. Round to the SOLUTION: nearest tenth. The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle. 17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm.

Therefore, the volume of the cone is about 235.6 in3.

So, the area of the base B is 24 cm2. 18.

Replace V with 144 and B with 24 in the formula for SOLUTION: the volume of a pyramid and solve for the height h. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 cm.

Find the volume of each cone. Round to the Therefore, the volume of the cone is about 134.8 3 nearest tenth. cm .

17. SOLUTION: 19.

The volume of a circular cone is , SOLUTION: where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Use a trigonometric ratio to find the height h of the cone.

Therefore, the volume of the cone is about 235.6 in3. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 1473.1 3 cone. The radius of this cone is 4.2 centimeters and cm . the height is 7.3 centimeters.

20. SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm .

Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the

cone. Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where 3 r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 2.8 ft . cone. The radius of this cone is 8 centimeters. 21. an oblique cone with a diameter of 16 inches and an

altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, Therefore, the volume of the cone is about 1473.1 the radius is or 8 inches. cm3.

20.

SOLUTION: Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant Use trigonometric ratios to find the height h and the height of 5.6 centimeters. Use the Pythagorean radius r of the cone. Theorem to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches 3 Therefore, the volume of the cone is about 5.8 cm . SOLUTION: 23. SNACKS Approximately how many cubic The volume of a circular cone is , where centimeters of roasted peanuts will completely fill a r is the radius of the base and h is the height of the paper cone that is 14 centimeters high and has a base cone. Since the diameter of this cone is 16 inches, diameter of 8 centimeters? Round to the nearest tenth. the radius is or 8 inches. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. Therefore, the volume of the cone is about 1072.3 in 3. 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Therefore, the paper cone will hold about 234.6 cm3 Theorem to find the height h of the cone. of roasted peanuts.

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the

greenhouse? 3 Therefore, the volume of the cone is about 5.8 cm .

SOLUTION: SOLUTION: The volume of a circular cone is , where The volume of a pyramid is , where B is the r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with centimeters, the radius is or 4 centimeters. The sides of 2 feet. A central angle of the octagon is height of the cone is 14 centimeters. or 45°, so the angle formed in the triangle below is 22.5°.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts. Use a trigonometric ratio to find the apothem a. 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid.

SOLUTION: The height of this pyramid is 5 feet.

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

Therefore, the volume of the greenhouse is about 25. GARDENING The greenhouse is a regular 3 octagonal pyramid with a height of 5 feet. The base 32.2 ft . has side lengths of 2 feet. What is the volume of the Find the volume of each solid. Round to the greenhouse? nearest tenth.

26. SOLUTION: SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Use a trigonometric ratio to find the apothem a.

27. SOLUTION:

The height of this pyramid is 5 feet.

Therefore, the volume of the greenhouse is about 3 32.2 ft . 28. Find the volume of each solid. Round to the SOLUTION: nearest tenth.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she

needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

27. SOLUTION:

The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 28. 6666.67 × 2 = 13,333 BTUs.

SOLUTION: 30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to It tells Marta how much clay is needed to make the purchase? model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. SOLUTION: c. Both the radius and the height are doubled. The building can be broken down into the rectangular SOLUTION: base and the pyramid ceiling. The volume of the base is Find the volume of the original cone. Then alter the values.

The volume of the ceiling is

a. Double h.

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. The volume is doubled.

b. Double r. 30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 2 The volume is multiplied by 2 or 4.

c. Double r and h.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius 3 of 4 centimeters and a height of 9 centimeters. volume is multiplied by 2 or 8. Describe how each change affects the volume of the cone. Find each measure. Round to the nearest tenth if necessary. a. The height is doubled. 32. A square pyramid has a volume of 862.5 cubic b. The radius is doubled. centimeters and a height of 11.5 centimeters. Find c. Both the radius and the height are doubled. the side length of the base. SOLUTION: SOLUTION: Find the volume of the original cone. Then alter the values. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled. The side length of the base is 15 cm.

b. Double r. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the 2 The volume is multiplied by 2 or 4. height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 c. Double r and h. centimeters.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find The diameter is 2(7) or 14 inches. the side length of the base. SOLUTION: 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the The volume of a pyramid is , where B is the volume of the cone? area of the base and h is the height of the pyramid. SOLUTION: Let s be the side length of the base.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter?

SOLUTION: So, the radius is about 3.8 millimeters.

The volume of a circular cone is , or Use the Pythagorean Theorem to find the height of the cone. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION: Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. The lateral area of a cone is , where r is c. ANALYTICAL Explain how multiplying the base the radius and is the slant height of the cone. area and/or the height of the pyramid by 5 affects the Replace L with 71.6 and with 6, then solve for the volume of the pyramid.

radius r. SOLUTION:

a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

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

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the b. The volumes are the same. The volume of a cone. pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the c. If the base area is multiplied by 5, the volume is two pyramids that you drew? Explain. multiplied by 5. If the height is multiplied by 5, the c. ANALYTICAL Explain how multiplying the base volume is multiplied by 5. If both the base area and area and/or the height of the pyramid by 5 affects the the height are multiplied by 5, the volume is multiplied volume of the pyramid. by 5 · 5 or 25. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h.

SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the

prism. Set the volumes equal.

The volumes will only be equal when the radius of the cone is equal to or when .

Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. SOLUTION: The volume of a cone with radius r and height h The slant height is used for surface area, but the equals the volume of a prism with height h. height is used for volume. For this cone, the slant SOLUTION: height of 13 is provided, and we need to calculate the height before we can calculate the volume. The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the Alexandra incorrectly used the slant height. prism. Set the volumes equal. 38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

OPEN ENDED 39. Give an example of a pyramid and a prism that have the same base and the same The volumes will only be equal when the radius of volume. Explain your reasoning. the cone is equal to or when . SOLUTION: Therefore, the statement is true sometimes if the The formula for volume of a prism is V = Bh and the base area of the cone is 3 times as great as the base formula for the volume of a pyramid is one-third of area of the prism. For example, if the base of the that. So, if a pyramid and prism have the same base, prism has an area of 10 square units, then its volume then in order to have the same volume, the height of is 10h cubic units. So, the cone must have a base the pyramid must be 3 times as great as the height of area of 30 square units so that its volume is the prism.

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast SOLUTION: finding volumes of pyramids and cones with finding The slant height is used for surface area, but the volumes of prisms and cylinders. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the SOLUTION: height before we can calculate the volume. To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has Alexandra incorrectly used the slant height. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the 38. REASONING A cone has a volume of 568 cubic same height and base area. centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain 41. A conical sand toy has the dimensions as shown your reasoning. below. How many cubic centimeters of sand will it hold when it is filled to the top? SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same A 12π radius and height. B 15π

39. OPEN ENDED Give an example of a pyramid and C a prism that have the same base and the same

volume. Explain your reasoning. D SOLUTION: The formula for volume of a prism is V = Bh and the SOLUTION: formula for the volume of a pyramid is one-third of Use the Pythagorean Theorem to find the radius r of that. So, if a pyramid and prism have the same base, the cone. then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

Sample answer:

A square pyramid with a base area of 16 and a So, the radius of the cone is 3 centimeters. height of 12, a prism with a square base area of 16 and a height of 4. The volume of a circular cone is , where 40. WRITING IN MATH Compare and contrast r is the radius of the base and h is the height of the finding volumes of pyramids and cones with finding cone. volumes of prisms and cylinders.

41. A conical sand toy has the dimensions as shown 42. SHORT RESPONSE Brooke is buying a tent that below. How many cubic centimeters of sand will it is in the shape of a rectangular pyramid. The base is hold when it is filled to the top? 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the A 12π area of the base and h is the height of the pyramid. B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

G So, the radius of the cone is 3 centimeters.

The volume of a circular cone is , where H

r is the radius of the base and h is the height of the J cone.

SOLUTION:

Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? eSolutions Manual - Powered by Cognero Page 16 SOLUTION: So, the correct choice is F. The volume of a pyramid is , where B is the SAT/ACT area of the base and h is the height of the pyramid. 44. For all A –8 B x – 4 C D E

SOLUTION:

Find the volume of each prism.

F 45. SOLUTION: G The volume of a prism is , where B is the area of the base and h is the height of the prism. The H base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the J prism is 6 inches.

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5 3 Therefore, the volume of the prism is 1008 in .

46. So, the correct choice is F. SOLUTION: The volume of a prism is , where B is the 44. SAT/ACT For all area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base A –8 of 10 feet and two legs of 13 feet. The height h will B x – 4 bisect the base. Use the Pythagorean Theorem to C find the height of the triangle. D E

SOLUTION:

So, the correct choice is E.

Find the volume of each prism. So, the height of the triangle is 12 feet. Find the area of the triangle.

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 2 inches and a width of 12 inches. The height h of the So, the area of the base B is 60 ft . prism is 6 inches. The height h of the prism is 19 feet.

3 3 Therefore, the volume of the prism is 1008 in . Therefore, the volume of the prism is 1140 ft .

46. 47. SOLUTION: SOLUTION: The volume of a prism is , where B is the The volume of a prism is , where B is the area of the base and h is the height of the prism. The area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base base of this prism is a rectangle with a length of 79.4 of 10 feet and two legs of 13 feet. The height h will meters and a width of 52.5 meters. The height of the bisect the base. Use the Pythagorean Theorem to prism is 102.3 meters. find the height of the triangle.

Therefore, the volume of the prism is about 426,437.6 3 m .

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write So, the height of the triangle is 12 feet. Find the area the exact answer and the answer rounded to the of the triangle. nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is

2 , where is r is the radius and l is the slant height So, the area of the base B is 60 ft . of the cone.

The height h of the prism is 19 feet. Find the slant height of the conical top.

3 Therefore, the volume of the prism is 1140 ft .

47. Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) SOLUTION: or 9 ft. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 3 m . The formula for finding the surface area of a cylinder 48. FARMING The picture shows a combination is , where is r is the radius and h is the slant

hopper cone and bin used by farmers to store grain height of the cylinder. after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write Surface area of the bin = Surface area of the the exact answer and the answer rounded to the cylinder + Surface area of the conical top + surface nearest square foot. area of the conical bottom. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height Find the area of each shaded region. Polygons of the cone. in 50 - 52 are regular.

Find the slant height of the conical top.

49.

SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

The formula for finding the surface area of a cylinder A regular hexagon has 6 sides, so the measure of the is , where is r is the radius and h is the slant height of the cylinder. interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon.

50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

51.

A regular hexagon has 6 sides, so the measure of the SOLUTION: The shaded area is the area of the equilateral triangle interior angle is . The apothem bisects the less the area of the inscribed circle. angle, so we use 30° when using trig to find the

length of the apothem. Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

51. So, the area of the equilateral triangle is about 67.3 SOLUTION: square feet. The shaded area is the area of the equilateral triangle

less the area of the inscribed circle. Subtract to find the area of shaded region.

Find the area of the circle with a radius of 3.6 feet.

Therefore, the area of the shaded region is about 2 So, the area of the circle is about 40.7 square feet. 26.6 ft .

Next, find the area of the equilateral triangle.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b The equilateral triangle can be divided into three represent the length of each side of the equilateral triangle and h represent the radius of the inscribed isosceles triangles with central angles of or circle. 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The Use a trigonometric ratio to find the value of b. angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find Therefore, the area of the shaded region is about the area of the shaded region. 2 26.6 ft . A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

52. SOLUTION:

The area of the shaded region is the area of the Therefore, the area of the shaded region is about circumscribed circle minus the area of the equilateral 2 triangle plus the area of the inscribed circle. Let b 168.2 mm . represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

2. SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters Find the volume of each cone. Round to the and a base 8 meters by 4.5 meters nearest tenth. SOLUTION:

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

6. SOLUTION:

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

Use trigonometry to find the radius r.

5. SOLUTION: The volume of a circular cone is , or

The volume of a circular cone is , or , where B is the area of the base, h is the height of the cone, and r is the radius of the base. , where B is the area of the base, h is the The height of the cone is 11.5 centimeters. height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. 6. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. SOLUTION:

Use trigonometry to find the radius r. 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: So, the height of the cone is 20 meters. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. 8. a cone with a slant height of 25 meters and a radius SOLUTION: of 15 meters The volume of a circular cone is , where B SOLUTION: is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

So, the height of the cone is 20 meters.

10. SOLUTION:

The volume of a pyramid is , where B is the 9. MUSEUMS The sky dome of the National Corvette area of the base and h is the height of the pyramid. Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the

base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet. 11.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION: 12. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume of a pyramid is , where B is the The length of the other leg of the right triangle is 6 area of the base and h is the height of the pyramid. cm.

Therefore, the height of the triangular pyramid is 18 cm.

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

17. SOLUTION: The length of the other leg of the right triangle is 6 cm. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

2 So, the area of the base B is 24 cm .

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

Therefore, the volume of the cone is about 235.6 in3.

17. SOLUTION:

19. SOLUTION:

Therefore, the volume of the cone is about 235.6 in3.

Use a trigonometric ratio to find the height h of the 18. cone. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and The volume of a circular cone is , where the height is 7.3 centimeters. r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Therefore, the volume of the cone is about 134.8 3 cm . Therefore, the volume of the cone is about 1473.1 cm3.

19. SOLUTION: 20. SOLUTION:

Use a trigonometric ratio to find the height h of the Use trigonometric ratios to find the height h and the cone. radius r of the cone.

Therefore, the volume of the cone is about 1473.1 cm3.

3 Therefore, the volume of the cone is about 2.8 ft .

Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1072.3 in 3.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 5.8 cm .

Therefore, the volume of the cone is about 1072.3 23. SNACKS Approximately how many cubic in 3. centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base 22. a right cone with a slant height of 5.6 centimeters diameter of 8 centimeters? Round to the nearest and a radius of 1 centimeter tenth. SOLUTION: SOLUTION: The cone has a radius r of 1 centimeter and a slant The volume of a circular cone is , where height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in

the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

SOLUTION:

Therefore, the paper cone will hold about 234.6 cm3 The volume of a pyramid is , where B is the

of roasted peanuts. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with 24. CCSS MODELING The Pyramid Arena in sides of 2 feet. A central angle of the octagon is Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its or 45°, so the angle formed in the triangle square base is 600 feet wide. Find the volume of this below is 22.5°. pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Use a trigonometric ratio to find the apothem a.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base The height of this pyramid is 5 feet. has side lengths of 2 feet. What is the volume of the greenhouse?

SOLUTION: Therefore, the volume of the greenhouse is about 3 The volume of a pyramid is , where B is the 32.2 ft . area of the base and h is the height of the pyramid. Find the volume of each solid. Round to the The base of the pyramid is a regular octagon with nearest tenth. sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet.

27.

Therefore, the volume of the greenhouse is about SOLUTION: 3 32.2 ft .

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

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

28.

SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

28.

SOLUTION: The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 29. HEATING Sam is building an art studio in her 6666.67 × 2 = 13,333 BTUs. backyard. To buy a heating unit for the space, she

needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction 30. SCIENCE Refer to page 825. Determine the with good insulation, there should be 2 BTUs per volume of the model. Explain why knowing the cubic foot. What size unit does Sam need to volume is helpful in this situation. purchase? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is It tells Marta how much clay is needed to make the model.

SCIENCE 30. Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. a. Double h. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume is doubled.

b. Double r.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. 2 The volume is multiplied by 2 or 4. a. The height is doubled. b. The radius is doubled. c. Double r and h. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find a. Double h. the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The volume is doubled.

b. Double r.

2 The volume is multiplied by 2 or 4.

c. Double r and h. The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

So, the radius is about 3.8 millimeters.

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

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters

and the slant height is 6 millimeters. What is the volume of the cone? So, the height of the cone is about 4.64 millimeters. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. So, the radius is about 3.8 millimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. Use the Pythagorean Theorem to find the height of c. ANALYTICAL Explain how multiplying the base the cone. area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this b. The volumes are the same. The volume of a problem, you will investigate rectangular pyramids. pyramid equals one third times the base area times a. GEOMETRIC Draw two pyramids with the height. So, if the base areas of two pyramids are different bases that have a height of 10 centimeters equal and their heights are equal, then their volumes and a base area of 24 square centimeters. are equal. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

c. If the base area is multiplied by 5, the volume is 36. CCSS ARGUMENTS Determine whether the multiplied by 5. If the height is multiplied by 5, the following statement is sometimes, always, or never volume is multiplied by 5. If both the base area and true. Justify your reasoning. the height are multiplied by 5, the volume is multiplied

by 5 · 5 or 25. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

or 10h cubic units. 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. 37. ERROR ANALYSIS Alexandra and Cornelio are The volume of a cone with radius r and height h calculating the volume of the cone below. Is either of equals the volume of a prism with height h. them correct? Explain your answer. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic The volumes will only be equal when the radius of centimeters. What is the volume of a cylinder that the cone is equal to or when . has the same radius and height as the cone? Explain Therefore, the statement is true sometimes if the your reasoning. base area of the cone is 3 times as great as the base SOLUTION: area of the prism. For example, if the base of the 3 prism has an area of 10 square units, then its volume 1704 cm ; The formula for the volume of a cylinder is 10h cubic units. So, the cone must have a base is V= Bh, while the formula for the volume of a cone area of 30 square units so that its volume is is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same or 10h cubic units. radius and height.

SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume. Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. Alexandra incorrectly used the slant height. Sample answer: 38. REASONING A cone has a volume of 568 cubic A square pyramid with a base area of 16 and a centimeters. What is the volume of a cylinder that height of 12, a prism with a square base area of 16 has the same radius and height as the cone? Explain and a height of 4. your reasoning. 40. WRITING IN MATH Compare and contrast SOLUTION: finding volumes of pyramids and cones with finding 1704 cm3; The formula for the volume of a cylinder volumes of prisms and cylinders. is V= Bh, while the formula for the volume of a cone SOLUTION: is V = Bh. The volume of a cylinder is three times To find the volume of each solid, you must know the as much as the volume of a cone with the same area of the base and the height. The volume of a radius and height. pyramid is one third the volume of a prism that has the same height and base area. The volume of a 39. OPEN ENDED Give an example of a pyramid and cone is one third the volume of a cylinder that has the a prism that have the same base and the same same height and base area. volume. Explain your reasoning. 41. A conical sand toy has the dimensions as shown SOLUTION: below. How many cubic centimeters of sand will it The formula for volume of a prism is V = Bh and the hold when it is filled to the top? formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism. A 12π B 15π

Set the base areas of the prism and pyramid, and SOLUTION: make the height of the pyramid equal to 3 times the Use the Pythagorean Theorem to find the radius r of

height of the prism. the cone.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION:

To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area. So, the radius of the cone is 3 centimeters. 41. A conical sand toy has the dimensions as shown The volume of a circular cone is , where below. How many cubic centimeters of sand will it hold when it is filled to the top? r is the radius of the base and h is the height of the cone.

A 12π B 15π

Therefore, the correct choice is A. D 42. SHORT RESPONSE Brooke is buying a tent that SOLUTION: is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, Use the Pythagorean Theorem to find the radius r of how tall is the tent’s center pole? the cone. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

G Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that H is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, J how tall is the tent’s center pole? SOLUTION: SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 The volume of a pyramid is , where B is the green} Number of possible outcomes : 25 area of the base and h is the height of the pyramid. Favorable outcomes: {5 orange} Number of favorable outcomes: 5

12-5 Volumes of Pyramids and Cones So, the correct choice is F.

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows 44. SAT/ACT For all the results of several spins. What is the experimental A –8 probability of the spinner landing on orange? B x – 4 C D

F SOLUTION:

So, the correct choice is F.

44. SAT/ACT For all

A –8 3 B x – 4 Therefore, the volume of the prism is 1008 in . C D E

SOLUTION: 46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will So, the correct choice is E. eSolutions Manual - Powered by Cognero bisect the base. Use the Pythagorean Theorem Pageto 17 Find the volume of each prism. find the height of the triangle.

45.

So, the height of the triangle is 12 feet. Find the area of the triangle.

3 Therefore, the volume of the prism is 1008 in .

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will 3 bisect the base. Use the Pythagorean Theorem to Therefore, the volume of the prism is 1140 ft . find the height of the triangle.

So, the height of the triangle is 12 feet. Find the area of the triangle.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain So, the area of the base B is 60 ft2. after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the The height h of the prism is 19 feet. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847.

47. Find the slant height of the conical top.

Therefore, the volume of the prism is about 426,437.6 Find the slant height of the conical bottom. 3 The height of the conical bottom is 28 – (5 + 12 + 2) m . or 9 ft.

48. FARMING The picture shows a combination

hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find The formula for finding the surface area of a cylinder the surface area of the cylinder and add them. is , where is r is the radius and h is the slant The formula for finding the surface area of a cone is height of the cylinder. , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top. Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) 49. or 9 ft. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder. 50. SOLUTION:

Area of the shaded region = Area of the circle – Surface area of the bin = Surface area of the Area of the hexagon cylinder + Surface area of the conical top + surface area of the conical bottom.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

Now find the area of the hexagon.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

51.

SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b. 51.

SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral Use a trigonometric ratio to find the value of b. triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the So, the area of the equilateral triangle is about 67.3 length of the side of the equilateral triangle as shown square feet. below.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about Use trigonometric ratios to find the values of b and 2 26.6 ft . h.

52.

SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b The area of the equilateral triangle will equal three represent the length of each side of the equilateral times the area of any of the isosceles triangles. Find triangle and h represent the radius of the inscribed the area of the shaded region. circle.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

Find the volume of each pyramid.

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters 1. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length The base of this pyramid is a right triangle with legs of 8 meters and a width of 4.5 meters. The height of of 9 inches and 5 inches and the height of the the pyramid is 5.2 meters.

pyramid is 10 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. 2.

SOLUTION:

5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius SOLUTION: is or 3.5 inches. The height of the cone is 4 inches. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

6. 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Find the volume of each cone. Round to the The height of the cone is 11.5 centimeters. nearest tenth.

5. SOLUTION: 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters The volume of a circular cone is , or SOLUTION:

8. a cone with a slant height of 25 meters and a radius of 15 meters

6. SOLUTION: SOLUTION:

Use the Pythagorean Theorem to find the height h of

Use trigonometry to find the radius r. the cone. Then find its volume.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. So, the height of the cone is 20 meters. The height of the cone is 11.5 centimeters.

7. an oblique cone with a height of 10.5 millimeters and 9. MUSEUMS The sky dome of the National Corvette a radius of 1.6 millimeters Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the SOLUTION: base is about 15,400 square feet, find the volume of The volume of a circular cone is , or air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. , where B is the area of the base, h is the SOLUTION:

height of the cone, and r is the radius of the base. The volume of a circular cone is , where B The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. 10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the height of the cone is 20 meters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the 11. base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have SOLUTION: to accommodate. Round to the nearest tenth. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a circular cone is , where B

is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

12. CCSS SENSE-MAKING Find the volume of SOLUTION: each pyramid. Round to the nearest tenth if necessary. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 11. 90° triangle.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the 15. a triangular pyramid with a height of 4.8 centimeters base and h is the height of the pyramid. and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters The base is a hexagon, so we need to make a right tri SOLUTION: determine the apothem. The interior angles of the he Find the height of the right triangle. 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the 15. a triangular pyramid with a height of 4.8 centimeters triangle. and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6

cm.

So, the area of the base B is 24 cm2.

The volume of a pyramid is , where B is the Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has Therefore, the height of the triangular pyramid is 18 a volume of 144 cubic centimeters. Find the height. cm.

SOLUTION: Find the volume of each cone. Round to the The base of the pyramid is a right triangle with a leg nearest tenth. of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle. 17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm.

Therefore, the volume of the cone is about 235.6 in3.

So, the area of the base B is 24 cm2.

18. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 cm.

Find the volume of each cone. Round to the nearest tenth. Therefore, the volume of the cone is about 134.8 3 cm .

17. SOLUTION: 19. The volume of a circular cone is , SOLUTION: where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Use a trigonometric ratio to find the height h of the cone.

Therefore, the volume of the cone is about 235.6 in3. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

18. SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and Therefore, the volume of the cone is about 1473.1 3 the height is 7.3 centimeters. cm .

20.

SOLUTION: Therefore, the volume of the cone is about 134.8 3 cm .

Use trigonometric ratios to find the height h and the radius r of the cone. 19. SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the Use a trigonometric ratio to find the height h of the cone. cone.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the 3 cone. The radius of this cone is 8 centimeters. Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 1473.1 cone. Since the diameter of this cone is 16 inches, 3 cm . the radius is or 8 inches.

20. SOLUTION: Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: Use trigonometric ratios to find the height h and the The cone has a radius r of 1 centimeter and a slant radius r of the cone. height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches 3 SOLUTION: Therefore, the volume of the cone is about 5.8 cm .

The volume of a circular cone is , where 23. SNACKS Approximately how many cubic r is the radius of the base and h is the height of the centimeters of roasted peanuts will completely fill a cone. Since the diameter of this cone is 16 inches, paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest the radius is or 8 inches. tenth.

SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The Therefore, the volume of the cone is about 1072.3 height of the cone is 14 centimeters. in 3.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base

has side lengths of 2 feet. What is the volume of the 3 Therefore, the volume of the cone is about 5.8 cm . greenhouse?

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION: The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the The volume of a pyramid is , where B is the cone. Since the diameter of the cone is 8 area of the base and h is the height of the pyramid. centimeters, the radius is or 4 centimeters. The The base of the pyramid is a regular octagon with height of the cone is 14 centimeters. sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

24. CCSS MODELING The Pyramid Arena in Use a trigonometric ratio to find the apothem a. Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid.

SOLUTION: The height of this pyramid is 5 feet. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular Therefore, the volume of the greenhouse is about octagonal pyramid with a height of 5 feet. The base 3 has side lengths of 2 feet. What is the volume of the 32.2 ft . greenhouse? Find the volume of each solid. Round to the nearest tenth.

26. SOLUTION: SOLUTION: Volume of the solid given = Volume of the small The volume of a pyramid is , where B is the cone + Volume of the large cone

area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Use a trigonometric ratio to find the apothem a.

27. SOLUTION: The height of this pyramid is 5 feet.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

Find the volume of each solid. Round to the 28. nearest tenth. SOLUTION:

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

29. HEATING Sam is building an art studio in her

backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is 27. SOLUTION:

The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic 28. foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. SOLUTION:

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase? It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. SOLUTION: b. The radius is doubled. The building can be broken down into the rectangular c. Both the radius and the height are doubled. base and the pyramid ceiling. The volume of the base SOLUTION: is Find the volume of the original cone. Then alter the values.

The volume of the ceiling is

a. Double h.

30. SCIENCE Refer to page 825. Determine the b. Double r. volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. 3 Describe how each change affects the volume of the volume is multiplied by 2 or 8. cone. Find each measure. Round to the nearest tenth a. The height is doubled. if necessary. b. The radius is doubled. 32. A square pyramid has a volume of 862.5 cubic c. Both the radius and the height are doubled. centimeters and a height of 11.5 centimeters. Find SOLUTION: the side length of the base. Find the volume of the original cone. Then alter the SOLUTION: values. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled. The side length of the base is 15 cm. b. Double r. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

2 , where B is the area of the base, h is the The volume is multiplied by 2 or 4. height of the cone, and r is the radius of the base. c. Double r and h. Since the diameter is 8 centimeters, the radius is 4 centimeters.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find

the side length of the base. The diameter is 2(7) or 14 inches. SOLUTION: 34. The lateral area of a cone is 71.6 square millimeters The volume of a pyramid is , where B is the and the slant height is 6 millimeters. What is the volume of the cone? area of the base and h is the height of the pyramid. Let s be the side length of the base. SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION: So, the radius is about 3.8 millimeters. The volume of a circular cone is , or Use the Pythagorean Theorem to find the height of , where B is the area of the base, h is the the cone.

height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION: Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the The lateral area of a cone is , where r is two pyramids that you drew? Explain. the radius and is the slant height of the cone. c. ANALYTICAL Explain how multiplying the base Replace L with 71.6 and with 6, then solve for the area and/or the height of the pyramid by 5 affects the radius r. volume of the pyramid. SOLUTION:

a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

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

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. If the base area is multiplied by 5, the volume is c. ANALYTICAL Explain how multiplying the base multiplied by 5. If the height is multiplied by 5, the area and/or the height of the pyramid by 5 affects the volume is multiplied by 5. If both the base area and volume of the pyramid. the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

The volumes will only be equal when the radius of

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h SOLUTION: equals the volume of a prism with height h. The slant height is used for surface area, but the SOLUTION: height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the The volume of a cone with a radius r and height h is height before we can calculate the volume. . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal. Alexandra incorrectly used the slant height.

39. OPEN ENDED Give an example of a pyramid and The volumes will only be equal when the radius of a prism that have the same base and the same the cone is equal to or when . volume. Explain your reasoning. Therefore, the statement is true sometimes if the SOLUTION: base area of the cone is 3 times as great as the base The formula for volume of a prism is V = Bh and the area of the prism. For example, if the base of the formula for the volume of a pyramid is one-third of prism has an area of 10 square units, then its volume that. So, if a pyramid and prism have the same base, is 10h cubic units. So, the cone must have a base then in order to have the same volume, the height of area of 30 square units so that its volume is the pyramid must be 3 times as great as the height of

or 10h cubic units. the prism.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4. SOLUTION: 40. WRITING IN MATH Compare and contrast The slant height is used for surface area, but the finding volumes of pyramids and cones with finding height is used for volume. For this cone, the slant volumes of prisms and cylinders. height of 13 is provided, and we need to calculate the height before we can calculate the volume. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a Alexandra incorrectly used the slant height. pyramid is one third the volume of a prism that has the same height and base area. The volume of a 38. REASONING A cone has a volume of 568 cubic cone is one third the volume of a cylinder that has the centimeters. What is the volume of a cylinder that same height and base area. has the same radius and height as the cone? Explain your reasoning. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it SOLUTION: hold when it is filled to the top? 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height. A 12π B 15π OPEN ENDED 39. Give an example of a pyramid and a prism that have the same base and the same C volume. Explain your reasoning.

SOLUTION: D The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of SOLUTION: that. So, if a pyramid and prism have the same base, Use the Pythagorean Theorem to find the radius r of then in order to have the same volume, the height of the cone. the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and

make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a

height of 12, a prism with a square base area of 16 So, the radius of the cone is 3 centimeters. and a height of 4. The volume of a circular cone is , where 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding r is the radius of the base and h is the height of the volumes of prisms and cylinders. cone.

SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area. Therefore, the correct choice is A. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it 42. SHORT RESPONSE Brooke is buying a tent that hold when it is filled to the top? is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

A 12π The volume of a pyramid is , where B is the B 15π area of the base and h is the height of the pyramid.

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters. G The volume of a circular cone is , where H r is the radius of the base and h is the height of the

cone. J

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5 Therefore, the correct choice is A.

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION: So, the correct choice is F. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. SAT/ACT 44. For all A –8 B x – 4 C D E

SOLUTION:

Find the volume of each prism.

F 45.

G SOLUTION: The volume of a prism is , where B is the H area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 J inches and a width of 12 inches. The height h of the prism is 6 inches. SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

3 Therefore, the volume of the prism is 1008 in .

So, the correct choice is F. 46. SOLUTION: 44. SAT/ACT For all The volume of a prism is , where B is the area of the base and h is the height of the prism. The A –8 base of this prism is an isosceles triangle with a base B x – 4 of 10 feet and two legs of 13 feet. The height h will C bisect the base. Use the Pythagorean Theorem to find the height of the triangle. D E

SOLUTION:

So, the correct choice is E.

Find the volume of each prism.

So, the height of the triangle is 12 feet. Find the area of the triangle.

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the 2 prism is 6 inches. So, the area of the base B is 60 ft .

The height h of the prism is 19 feet.

12-5 Volumes of Pyramids and Cones 3 Therefore, the volume of the prism is 1008 in . 3 Therefore, the volume of the prism is 1140 ft .

46. 47. SOLUTION: The volume of a prism is , where B is the SOLUTION: area of the base and h is the height of the prism. The The volume of a prism is , where B is the base of this prism is an isosceles triangle with a base area of the base and h is the height of the prism. The of 10 feet and two legs of 13 feet. The height h will base of this prism is a rectangle with a length of 79.4 bisect the base. Use the Pythagorean Theorem to meters and a width of 52.5 meters. The height of the find the height of the triangle. prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface So, the height of the triangle is 12 feet. Find the area area of the bin with a conical top and bottom. Write of the triangle. the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is 2 So, the area of the base B is 60 ft . , where is r is the radius and l is the slant height of the cone. The height h of the prism is 19 feet. Find the slant height of the conical top.

3 Therefore, the volume of the prism is 1140 ft .

eSolutions47. Manual - Powered by Cognero Page 18 Find the slant height of the conical bottom. SOLUTION: The height of the conical bottom is 28 – (5 + 12 + 2) The volume of a prism is , where B is the or 9 ft. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 3 m . The formula for finding the surface area of a cylinder 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain is , where is r is the radius and h is the slant

after harvest. The cone at the bottom of the bin height of the cylinder. allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the Surface area of the bin = Surface area of the nearest square foot. cylinder + Surface area of the conical top + surface Refer to the photo on Page 847. area of the conical bottom.

SOLUTION:

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant A regular hexagon has 6 sides, so the measure of the height of the cylinder. interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49.

SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

51. A regular hexagon has 6 sides, so the measure of the SOLUTION: interior angle is . The apothem bisects the The shaded area is the area of the equilateral triangle angle, so we use 30° when using trig to find the less the area of the inscribed circle. length of the apothem.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

51. SOLUTION: So, the area of the equilateral triangle is about 67.3 The shaded area is the area of the equilateral triangle square feet. less the area of the inscribed circle. Subtract to find the area of shaded region. Find the area of the circle with a radius of 3.6 feet.

Therefore, the area of the shaded region is about So, the area of the circle is about 40.7 square feet. 2 26.6 ft . Next, find the area of the equilateral triangle.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral The equilateral triangle can be divided into three triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral isosceles triangles with central angles of or triangle and h represent the radius of the inscribed 120°, so the angle in the triangle created by the circle. height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Divide the equilateral triangle into three isosceles Use a trigonometric ratio to find the value of b. triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

The area of the equilateral triangle will equal three Therefore, the area of the shaded region is about times the area of any of the isosceles triangles. Find 2 26.6 ft . the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

52. SOLUTION:

The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral Therefore, the area of the shaded region is about triangle plus the area of the inscribed circle. Let b 2 represent the length of each side of the equilateral 168.2 mm . triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

Find the volume of each pyramid. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

2. SOLUTION:

SOLUTION:

The volume of a circular cone is , or

and a base 8 meters by 4.5 meters SOLUTION:

6. SOLUTION:

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

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

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters 5. SOLUTION:

SOLUTION: The volume of a circular cone is , or The volume of a circular cone is , or , where B is the area of the base, h is the , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height of the cone, and r is the radius of the base. height is 10.5 millimeters. Since the diameter of this cone is 7 inches, the radius

is or 3.5 inches. The height of the cone is 4 inches.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

6. SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

Use trigonometry to find the radius r.

The volume of a circular cone is , or So, the height of the cone is 20 meters.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of 7. an oblique cone with a height of 10.5 millimeters and air that the heating and cooling systems would have a radius of 1.6 millimeters to accommodate. Round to the nearest tenth. SOLUTION: SOLUTION:

CCSS SENSE-MAKING Find the volume of 8. a cone with a slant height of 25 meters and a radius each pyramid. Round to the nearest tenth if of 15 meters necessary. SOLUTION:

10. SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

So, the height of the cone is 20 meters.

11.

9. MUSEUMS The sky dome of the National Corvette SOLUTION: Museum in Bowling Green, Kentucky, is a conical The volume of a pyramid is , where B is the building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of area of the base and h is the height of the pyramid. air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if

necessary.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 13.

SOLUTION: The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square area of the base and h is the height of the pyramid. feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and SOLUTION: hypotenuse 10.2 centimeters

The volume of a pyramid is , where B is the SOLUTION: Find the height of the right triangle. base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the height of the triangular pyramid is 18 cm.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 3 cm. Therefore, the volume of the cone is about 235.6 in .

Therefore, the volume of the cone is about 134.8 Therefore, the height of the triangular pyramid is 18 3 cm. cm .

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

19. 17. SOLUTION: SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 Use a trigonometric ratio to find the height h of the cone.

inches.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. 3 Therefore, the volume of the cone is about 235.6 in .

18. SOLUTION: Therefore, the volume of the cone is about 1473.1 cm3. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

20. SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm . Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

The volume of a circular cone is , where 21. an oblique cone with a diameter of 16 inches and an

r is the radius of the base and h is the height of the altitude of 16 inches cone. The radius of this cone is 8 centimeters. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1473.1 cm3.

Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft . 3 Therefore, the volume of the cone is about 5.8 cm . 21. an oblique cone with a diameter of 16 inches and an SNACKS altitude of 16 inches 23. Approximately how many cubic centimeters of roasted peanuts will completely fill a SOLUTION: paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest The volume of a circular cone is , where tenth. r is the radius of the base and h is the height of the SOLUTION: cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Therefore, the volume of the cone is about 1072.3 in 3.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

3 Therefore, the volume of the cone is about 5.8 cm .

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest SOLUTION: tenth. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a circular cone is , where The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 or 45°, so the angle formed in the triangle below is 22.5°. centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Use a trigonometric ratio to find the apothem a. Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

Find the volume of each solid. Round to the GARDENING 25. The greenhouse is a regular nearest tenth. octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

SOLUTION:

Use a trigonometric ratio to find the apothem a. 27.

SOLUTION:

The height of this pyramid is 5 feet.

28. Therefore, the volume of the greenhouse is about SOLUTION: 3 32.2 ft .

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

26. SOLUTION: 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she Volume of the solid given = Volume of the small needs to determine the BTUs (British Thermal Units) cone + Volume of the large cone required to heat the building. For new construction

with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

27.

SOLUTION: The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the

28. volume of the model. Explain why knowing the SOLUTION: volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The building can be broken down into the rectangular values. base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is a. Double h.

The total volume is therefore 5000 + 1666.67 = The volume is doubled. 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is b. Double r. 6666.67 × 2 = 13,333 BTUs.

area of the base and h is the height of the pyramid. c. Double r and h.

It tells Marta how much clay is needed to make the 3 model. volume is multiplied by 2 or 8. Find each measure. Round to the nearest tenth 31. CHANGING DIMENSIONS A cone has a radius if necessary. of 4 centimeters and a height of 9 centimeters. 32. A square pyramid has a volume of 862.5 cubic Describe how each change affects the volume of the centimeters and a height of 11.5 centimeters. Find cone. the side length of the base. a. The height is doubled. b. The radius is doubled. SOLUTION: c. Both the radius and the height are doubled. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. Find the volume of the original cone. Then alter the Let s be the side length of the base.

values.

a. Double h.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the

The volume is doubled. height is 12 inches. What is the diameter?

b. Double r. SOLUTION: The volume of a circular cone is , or

c. Double r and h.

volume is multiplied by 23 or 8.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The side length of the base is 15 cm.

So, the radius is about 3.8 millimeters. 33. The volume of a cone is 196π cubic inches and the

height is 12 inches. What is the diameter? Use the Pythagorean Theorem to find the height of SOLUTION: the cone.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters Therefore, the volume of the cone is about 70.2 and the slant height is 6 millimeters. What is the 3 volume of the cone? mm . SOLUTION: 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. The lateral area of a cone is , where r is SOLUTION: the radius and is the slant height of the cone. a. Replace L with 71.6 and with 6, then solve for the Use rectangular bases and pick values that

radius r. multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

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

Therefore, the volume of the cone is about 70.2 3 mm .

Sample answer:

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning.

or 10h cubic units.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: SOLUTION: The slant height is used for surface area, but the To find the volume of each solid, you must know the height is used for volume. For this cone, the slant area of the base and the height. The volume of a height of 13 is provided, and we need to calculate the pyramid is one third the volume of a prism that has height before we can calculate the volume. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area. Alexandra incorrectly used the slant height. 41. A conical sand toy has the dimensions as shown 38. REASONING A cone has a volume of 568 cubic below. How many cubic centimeters of sand will it centimeters. What is the volume of a cylinder that hold when it is filled to the top? has the same radius and height as the cone? Explain your reasoning. SOLUTION: 1704 cm3; The formula for the volume of a cylinder A 12π is V= Bh, while the formula for the volume of a cone B 15π is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same C radius and height.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. So, the radius of the cone is 3 centimeters.

A 12π B 15π

green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

Therefore, the correct choice is A.

The volume of a pyramid is , where B is the A –8 B x – 4 area of the base and h is the height of the pyramid. C

D E

SOLUTION:

45. SOLUTION:

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The

So, the correct choice is F. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will 44. SAT/ACT For all bisect the base. Use the Pythagorean Theorem to find the height of the triangle. A –8 B x – 4 C D

SOLUTION:

So, the correct choice is E. So, the height of the triangle is 12 feet. Find the area of the triangle. Find the volume of each prism.

3 Therefore, the volume of the prism is 1140 ft .

3 Therefore, the volume of the prism is 1008 in .

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. So, the height of the triangle is 12 feet. Find the area SOLUTION: of the triangle. To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

2 Find the slant height of the conical top. So, the area of the base B is 60 ft .

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1140 ft .

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 47.

The formula for finding the surface area of a cylinder Therefore, the volume of the prism is about 426,437.6 is , where is r is the radius and h is the slant 3 height of the cylinder. m . 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin Surface area of the bin = Surface area of the allows the grain to be emptied more easily. Use the cylinder + Surface area of the conical top + surface dimensions in the diagram to find the entire surface area of the conical bottom. area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find Find the area of each shaded region. Polygons the surface area of the cylinder and add them. in 50 - 52 are regular. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

49. Find the slant height of the conical top. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

12-5 Volumes of Pyramids and Cones

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50 Now find the area of the hexagon.

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50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. A regular hexagon has 6 sides, so the measure of the Find the area of the circle with a radius of 3.6 feet. interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three

Use a trigonometric ratio to find the value of b. Now find the area of the hexagon.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet. 51. Subtract to find the area of shaded region. SOLUTION:

The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 2 26.6 ft .

So, the area of the circle is about 40.7 square feet.

52. Next, find the area of the equilateral triangle. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Now use the area of the isosceles triangles to find Use trigonometric ratios to find the values of b and the area of the equilateral triangle. h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A Therefore, the area of the shaded region is about (equilateral triangle) + A(inscribed circle) 2 26.6 ft .

52. Therefore, the area of the shaded region is about 2 SOLUTION: 168.2 mm . The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters Find the volume of each pyramid. and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the

1. area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length SOLUTION: of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

2. SOLUTION: Find the volume of each cone. Round to the The volume of a pyramid is , where B is the nearest tenth. area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12

centimeters. 5.

SOLUTION: The volume of a circular cone is , or

6. SOLUTION:

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths

SOLUTION: Use trigonometry to find the radius r.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 The volume of a circular cone is , or meters. The height of the pyramid is 14 meters.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

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

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: 5. SOLUTION: The volume of a circular cone is , or

The volume of a circular cone is , or , where B is the area of the base, h is the height of the cone, and r is the radius of the base. , where B is the area of the base, h is the The radius of this cone is 1.6 millimeters and the height of the cone, and r is the radius of the base. height is 10.5 millimeters. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

SOLUTION: Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

Use trigonometry to find the radius r.

So, the height of the cone is 20 meters. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have 7. an oblique cone with a height of 10.5 millimeters and to accommodate. Round to the nearest tenth. a radius of 1.6 millimeters SOLUTION: SOLUTION: The volume of a circular cone is , where B The volume of a circular cone is , or is the area of the base and h is the height of the , where B is the area of the base, h is the cone. For this cone, the area of the base is 15,400 square height of the cone, and r is the radius of the base. feet and the height is 100 feet. The radius of this cone is 1.6 millimeters and the

height is 10.5 millimeters.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if 8. a cone with a slant height of 25 meters and a radius necessary. of 15 meters SOLUTION:

10. SOLUTION:

Use the Pythagorean Theorem to find the height h of The volume of a pyramid is , where B is the the cone. Then find its volume. area of the base and h is the height of the pyramid.

So, the height of the cone is 20 meters.

11. SOLUTION: 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical The volume of a pyramid is , where B is the building. If the height is 100 feet and the area of the area of the base and h is the height of the pyramid. base is about 15,400 square feet, find the volume of

air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION:

The volume of a pyramid is , where B is the 13. area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the height of the triangular pyramid is 18 cm.

of 8 centimeters and a hypotenuse of 10 centimeters. 17. Use the Pythagorean Theorem to find the other leg a SOLUTION: of the right triangle and then find the area of the triangle. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 Therefore, the volume of the cone is about 235.6 in3. cm.

18.

SOLUTION: 2 So, the area of the base B is 24 cm . The volume of a circular cone is , where

Therefore, the volume of the cone is about 134.8 3 Therefore, the height of the triangular pyramid is 18 cm . cm.

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

19. SOLUTION: 17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the Use a trigonometric ratio to find the height h of the radius is or 5 inches. The height of the cone is 9 cone. inches.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. Therefore, the volume of the cone is about 235.6 in3.

SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm . Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an The volume of a circular cone is , where altitude of 16 inches r is the radius of the base and h is the height of the SOLUTION: cone. The radius of this cone is 8 centimeters. The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1473.1 cm3.

Therefore, the volume of the cone is about 1072.3 in 3. 20. 22. a right cone with a slant height of 5.6 centimeters SOLUTION: and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 3 Therefore, the volume of the cone is about 2.8 ft . Therefore, the volume of the cone is about 5.8 cm .

21. an oblique cone with a diameter of 16 inches and an 23. SNACKS Approximately how many cubic altitude of 16 inches centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base SOLUTION: diameter of 8 centimeters? Round to the nearest tenth. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, The volume of a circular cone is , where the radius is or 8 inches. r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8

centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Therefore, the volume of the cone is about 1072.3 in 3.

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter Therefore, the paper cone will hold about 234.6 cm3 SOLUTION: of roasted peanuts. The cone has a radius r of 1 centimeter and a slant 24. CCSS MODELING The Pyramid Arena in height of 5.6 centimeters. Use the Pythagorean Memphis, Tennessee, is the third largest pyramid in Theorem to find the height h of the cone. the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

3 Therefore, the volume of the cone is about 5.8 cm .

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base SOLUTION: diameter of 8 centimeters? Round to the nearest tenth. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with The volume of a circular cone is , where sides of 2 feet. A central angle of the octagon is r is the radius of the base and h is the height of the or 45°, so the angle formed in the triangle cone. Since the diameter of the cone is 8 below is 22.5°. centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

Use a trigonometric ratio to find the apothem a.

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

SOLUTION:

Use a trigonometric ratio to find the apothem a. 27. SOLUTION:

The height of this pyramid is 5 feet.

28.

Therefore, the volume of the greenhouse is about SOLUTION: 3 32.2 ft .

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

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

27.

SOLUTION: The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the 28. volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she It tells Marta how much clay is needed to make the needs to determine the BTUs (British Thermal Units) model. required to heat the building. For new construction with good insulation, there should be 2 BTUs per 31. CHANGING DIMENSIONS A cone has a radius cubic foot. What size unit does Sam need to of 4 centimeters and a height of 9 centimeters. purchase? Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION:

Find the volume of the original cone. Then alter the SOLUTION: values. The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

a. Double h. The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = The volume is doubled.

6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is b. Double r. 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation.

SOLUTION: 2 The volume is multiplied by 2 or 4. The volume of a pyramid is , where B is the c. Double r and h. area of the base and h is the height of the pyramid.

3 It tells Marta how much clay is needed to make the volume is multiplied by 2 or 8. model. Find each measure. Round to the nearest tenth 31. CHANGING DIMENSIONS A cone has a radius if necessary. of 4 centimeters and a height of 9 centimeters. 32. A square pyramid has a volume of 862.5 cubic Describe how each change affects the volume of the centimeters and a height of 11.5 centimeters. Find cone. the side length of the base. a. The height is doubled. SOLUTION: b. The radius is doubled. c. Both the radius and the height are doubled. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. Find the volume of the original cone. Then alter the Let s be the side length of the base. values.

a. Double h.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the The volume is doubled. height is 12 inches. What is the diameter? SOLUTION: b. Double r. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

3 volume is multiplied by 2 or 8. Find each measure. Round to the nearest tenth The diameter is 2(7) or 14 inches. if necessary. 34. The lateral area of a cone is 71.6 square millimeters 32. A square pyramid has a volume of 862.5 cubic and the slant height is 6 millimeters. What is the centimeters and a height of 11.5 centimeters. Find volume of the cone? the side length of the base. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches. 34. The lateral area of a cone is 71.6 square millimeters Therefore, the volume of the cone is about 70.2 3 and the slant height is 6 millimeters. What is the mm . volume of the cone? 35. MULTIPLE REPRESENTATIONS In this SOLUTION: problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid.

So, the radius is about 3.8 millimeters.

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

So, the height of the cone is about 4.64 millimeters. b. The volumes are the same. The volume of a pyramid equals one third times the base area times The volume of a circular cone is , where the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes r is the radius of the base and h is the height of the are equal. cone.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this

problem, you will investigate rectangular pyramids. c. a. GEOMETRIC Draw two pyramids with If the base area is multiplied by 5, the volume is different bases that have a height of 10 centimeters multiplied by 5. If the height is multiplied by 5, the and a base area of 24 square centimeters. volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied b. VERBAL What is true about the volumes of the by 5 · 5 or 25. two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

SOLUTION: The slant height is used for surface area, but the 36. CCSS ARGUMENTS Determine whether the height is used for volume. For this cone, the slant following statement is sometimes, always, or never height of 13 is provided, and we need to calculate the true. Justify your reasoning. height before we can calculate the volume.

The volume of a cone with radius r and height h

equals the volume of a prism with height h. SOLUTION: Alexandra incorrectly used the slant height. The volume of a cone with a radius r and height h is 38. REASONING A cone has a volume of 568 cubic . The volume of a prism with a height of centimeters. What is the volume of a cylinder that h is where B is the area of the base of the has the same radius and height as the cone? Explain prism. Set the volumes equal. your reasoning.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION:

The formula for volume of a prism is V = Bh and the The volumes will only be equal when the radius of formula for the volume of a pyramid is one-third of the cone is equal to or when . that. So, if a pyramid and prism have the same base, Therefore, the statement is true sometimes if the then in order to have the same volume, the height of base area of the cone is 3 times as great as the base the pyramid must be 3 times as great as the height of

area of the prism. For example, if the base of the the prism.

prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

ERROR ANALYSIS 37. Alexandra and Cornelio are Set the base areas of the prism and pyramid, and calculating the volume of the cone below. Is either of make the height of the pyramid equal to 3 times the

them correct? Explain your answer. height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16

and a height of 4. 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders.

Alexandra incorrectly used the slant height. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it 38. REASONING A cone has a volume of 568 cubic hold when it is filled to the top? centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning.

SOLUTION: 1704 cm3; The formula for the volume of a cylinder A 12π is V= Bh, while the formula for the volume of a cone B 15π

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. So, the radius of the cone is 3 centimeters.

The volume of a circular cone is , where Sample answer: A square pyramid with a base area of 16 and a r is the radius of the base and h is the height of the height of 12, a prism with a square base area of 16 cone. and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a Therefore, the correct choice is A. pyramid is one third the volume of a prism that has the same height and base area. The volume of a 42. SHORT RESPONSE Brooke is buying a tent that cone is one third the volume of a cylinder that has the is in the shape of a rectangular pyramid. The base is same height and base area. 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it SOLUTION: hold when it is filled to the top? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

A 12π B 15π

So, the radius of the cone is 3 centimeters. J

The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the Possible outcomes: {6 red, 4 blue, 5 orange, 10 cone. green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

Therefore, the correct choice is A.

SOLUTION:

45. SOLUTION:

The volume of a prism is , where B is the F area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14

G inches and a width of 12 inches. The height h of the prism is 6 inches.

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 3 green} Therefore, the volume of the prism is 1008 in . Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The So, the correct choice is F. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to 44. SAT/ACT For all find the height of the triangle. A –8 B x – 4 C D E

SOLUTION:

So, the height of the triangle is 12 feet. Find the area So, the correct choice is E. of the triangle.

Find the volume of each prism.

45.

prism is 6 inches.

3 Therefore, the volume of the prism is 1140 ft .

3 Therefore, the volume of the prism is 1008 in .

47. SOLUTION: The volume of a prism is , where B is the 46. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 SOLUTION: meters and a width of 52.5 meters. The height of the The volume of a prism is , where B is the prism is 102.3 meters. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847.

Find the slant height of the conical top. So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1140 ft .

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant Therefore, the volume of the prism is about 426,437.6 height of the cylinder. m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain Surface area of the bin = Surface area of the after harvest. The cone at the bottom of the bin cylinder + Surface area of the conical top + surface allows the grain to be emptied more easily. Use the area of the conical bottom. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the Find the area of each shaded region. Polygons surface area of the conical top and bottom and find in 50 - 52 are regular. the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. 49.

SOLUTION: Find the slant height of the conical top. Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50 Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon 51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

Now find the area of the hexagon.

12-5 Volumes of Pyramids and Cones Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet.

51. Subtract to find the area of shaded region. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 2 26.6 ft .

So, the area of the circle is about 40.7 square feet. 52.

Next, find the area of the equilateral triangle. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Now use the area of the isosceles triangles to find Use trigonometric ratios to find the values of b and the area of the equilateral triangle. h. eSolutions Manual - Powered by Cognero Page 20

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle) Therefore, the area of the shaded region is about 2 26.6 ft .

Therefore, the area of the shaded region is about 52. 168.2 mm2. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

Find the volume of each pyramid.

1. SOLUTION:

2. SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

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

5. SOLUTION:

The volume of a circular cone is , or

6. SOLUTION:

Use trigonometry to find the radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

So, the height of the cone is 20 meters.

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

Therefore, the height of the triangular pyramid is 18 cm.

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

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Therefore, the volume of the cone is about 235.6 in3.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 134.8 3 cm .

19. SOLUTION:

Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Therefore, the volume of the cone is about 1473.1 cm3.

20. SOLUTION:

Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 in 3.

3 Therefore, the volume of the cone is about 5.8 cm .

Therefore, the paper cone will hold about 234.6 cm3 of roasted peanuts.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

SOLUTION:

Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet.

Therefore, the volume of the greenhouse is about 3 32.2 ft .

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

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

27. SOLUTION:

28. SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 6666.67 ft3. Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

30. SCIENCE Refer to page 825. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

a. Double h.

The volume is doubled.

b. Double r.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The side length of the base is 15 cm.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The diameter is 2(7) or 14 inches.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

So, the radius is about 3.8 millimeters.

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

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

Alexandra incorrectly used the slant height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the correct choice is A.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

So, the correct choice is F.

44. SAT/ACT For all

A –8 B x – 4 C D E

SOLUTION:

So, the correct choice is E.

Find the volume of each prism.

3 Therefore, the volume of the prism is 1008 in .

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the height of the triangle is 12 feet. Find the area of the triangle.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1140 ft .

Therefore, the volume of the prism is about 426,437.6 m3.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 847. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 26.6 ft .

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

12-5 Volumes of Pyramids and Cones Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.