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Find the volume of each pyramid.

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 legsof 9 inches and 5 inches and the height of thepyramid is 10 inches.

ANSWER:

75 in3

2.

SOLUTION:


area of the base and h is the height of the pyramid.The base of this pyramid is a regular pentagon withsides of 4.4 centimeters and an apothem of 3centimeters. The height of the pyramid is 12centimeters.

ANSWER:

132 cm3

3. a rectangular pyramid with a height of 5.2 meters anda 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 rectangle with a lengthof 8 meters and a width of 4.5 meters. The height ofthe pyramid is 5.2 meters.

ANSWER:

62.4 m3

eSolutions Manual - Powered by Cognero Page 1

11-3 Volumes of Pyramids and Cones

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

SOLUTION:


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

ANSWER:

298.7 m3

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

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.

ANSWER:

51.3 in3

6.

SOLUTION:

Use trigonometry to find the radius r.



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

ANSWER:

168.1 cm3



7. an oblique cone with a height of 10.5 millimeters anda radius of 1.6 millimeters

SOLUTION:



height of the cone, and r is the radius of the base.The radius of this cone is 1.6 millimeters and theheight is 10.5 millimeters.

ANSWER:

28.1 mm3

8. a cone with a slant height of 25 meters and a radiusof 15 meters

SOLUTION:

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

So, the height of the cone is 20 meters.

ANSWER:

4712.4 m3



9. HUTS The Caddo Indians, from the east part ofTexas, lived in tall cone-shaped grass huts made ofwooden pole frames with long prairie grasses driedand threaded through the poles in layers. Thesehouses normally measured 40 feet tall. Suppose thediameter is also 40 feet. What is the volume insidethe hut?

SOLUTION:

The volume of a circular cone is , where B

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

ANSWER:

about 16,755 ft3

STRUCTURE Find the volume of each pyramid.Round to the nearest tenth.

10.

SOLUTION:


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

ANSWER:

605 in3

11.

SOLUTION:



ANSWER:

105.8 mm3



12.

SOLUTION:



ANSWER:

482.1 m3

13.

SOLUTION:


area of the base and h is the height of the pyramid. The base is a hexagon, so we need to make a righttriangle to determine the apothem. The interior anglesof the hexagon are 120°. The radius bisects the angle,so the right triangle is a 30°-60°-90° triangle.

The apothem is .

ANSWER:

233.8 cm3

14. a pentagonal pyramid with a base area of 590 squarefeet and an altitude of 7 feet

SOLUTION:



ANSWER:

1376.7 ft3



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

SOLUTION:

Find the height of the right triangle.



ANSWER:

35.6 cm3

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

SOLUTION:

The base of the pyramid is a right triangle with a legof 8 centimeters and a hypotenuse of 10 centimeters.

Use the Pythagorean Theorem to find the other leg aof the right triangle and then find the area of thetriangle.

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

So, the area of the base B is 24 cm2. Replace V with 144 and B with 24 in the formula forthe volume of a pyramid and solve for the height h.

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

ANSWER:

18 cm



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

17.

SOLUTION:

The volume of a circular cone is ,

where r is the radius of the base and h is the heightof 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.

ANSWER:

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 thecone. The radius of this cone is 4.2 centimeters andthe height is 7.3 centimeters.

Therefore, the volume of the cone is about 134.8

cm3.

ANSWER:

134.8 cm3



19.

SOLUTION:

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


r is the radius of the base and h is the height of thecone. The radius of this cone is 8 centimeters.


cm3.

ANSWER:

1473.1 cm3

20.

SOLUTION:

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


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

Therefore, the volume of the cone is about 2.8 ft3.

ANSWER:

2.8 ft3



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

SOLUTION:


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

radius is or 8 inches.


in3.

ANSWER:

1072.3 in3

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

SOLUTION:

The cone has a radius r of 1 centimeter and a slantheight of 5.6 centimeters. Use the PythagoreanTheorem to find the height h of the cone.

Therefore, the volume of the cone is about 5.8 cm3.

ANSWER:

5.8 cm3



23. SNACKS Approximately how many cubiccentimeters of roasted peanuts will completely fill apaper cone that is 14 centimeters high and has a basediameter of 8 centimeters? Round to the nearesttenth.

SOLUTION:


r is the radius of the base and h is the height of thecone. Since the diameter of the cone is 8 centimeters,

the radius is or 4 centimeters. The height of thecone is 14 centimeters.

Therefore, the paper cone will hold about 234.6 cm3

of roasted peanuts.

ANSWER:

234.6 cm3

24. MODELING A pyramid-shaped building inMemphis, Tennessee, is approximately 350 feet tall,and its square base is 600 feet wide. Find the volumeof this pyramid.

SOLUTION:



ANSWER:

42,000,000 ft3

25. GARDENING The greenhouse is a regular

octagonal pyramid with a height of 5 feet. The basehas side lengths of 2 feet. What is the volume of thegreenhouse?

SOLUTION:


area of the base and h is the height of the pyramid.The base of the pyramid is a regular octagon withsides 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.

Therefore, the volume of the greenhouse is about

32.2 ft3.



ANSWER:

32.2 ft3

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

26.

SOLUTION:

Volume of the solid given = Volume of the small cone+ Volume of the large cone

ANSWER:

471.2 in3

27.

SOLUTION:

ANSWER:

3190.6 m3

28.

SOLUTION:

ANSWER:

7698.5 cm3



29. HEATING Sam is building an art studio in herbackyard. To buy a heating unit for the space, sheneeds to determine the BTUs (British Thermal Units)required to heat the building. For new constructionwith good insulation, there should be 2 BTUs percubic foot. What size unit does Sam need topurchase?

SOLUTION: The building can be broken down into the rectangularbase and the pyramid ceiling. The volume of the baseis

The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 =

6666.67 ft3. Two BTU's are needed for every cubicfoot, so the size of the heating unit Sam should buy is6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Refer to page 810. Determine thevolume of the crystal model that Marta is making.Explain why knowing the volume is helpful in thissituation.

SOLUTION:



It tells Marta how much clay is needed to make themodel.

ANSWER:

2 in3; It tells Marta how much clay is needed to makethe model.

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

a. Double h.



The volume is doubled. b. Double r.

The volume is multiplied by 22 or 4. c. Double r and h.

volume is multiplied by 23 or 8.

ANSWER:

a. The volume is doubled.b. The volume is multiplied by 22 or 4.c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenthif necessary.

32. A square pyramid has a volume of 862.5 cubiccentimeters and a height of 11.5 centimeters. Find theside length of the base.

SOLUTION:


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.

ANSWER:

15 cm



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

SOLUTION:



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

The diameter is 2(7) or 14 inches.

ANSWER:

14 in.

34. The lateral area of a cone is 71.6 square millimetersand the slant height is 6 millimeters. What is thevolume of the cone?

SOLUTION:

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

So, the radius is about 3.8 millimeters. Use the Pythagorean Theorem to find the height ofthe cone.

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


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

Therefore, the volume of the cone is about 70.2 mm3.

ANSWER:

70.2 mm3

35. MULTIPLE REPRESENTATIONS In thisproblem, you will investigate rectangular pyramids.a. GEOMETRIC Draw two pyramids with differentbases that have a height of 10 centimeters and a basearea of 24 square centimeters.b. VERBAL What is true about the volumes of thetwo pyramids that you drew? Explain.c. ANALYTICAL Explain how multiplying the basearea and/or the height of the pyramid by 5 affects thevolume of the pyramid.



SOLUTION:

a. Use rectangular bases and pick values that multiplyto make 24. Sample answer:

b. The volumes are the same. The volume of apyramid equals one third times the base area timesthe height. So, if the base areas of two pyramids areequal and their heights are equal, then their volumesare equal.

c. If the base area is multiplied by 5, the volume ismultiplied by 5. If the height is multiplied by 5, thevolume is multiplied by 5. If both the base area andthe height are multiplied by 5, the volume is multipliedby 5 · 5 or 25.

ANSWER:

a. Sample answer:



b. The volumes are the same. The volume of apyramid equals one third times the base area timesthe height. So, if the base areas of two pyramids areequal and their heights are equal, then their volumesare equal.c. If the base area is multiplied by 5, the volume ismultiplied by 5. If the height is multiplied by 5, thevolume is multiplied by 5. If both the base area andthe height are multiplied by 5, the volume is multipliedby 5 · 5 or 25.

36. CONSTRUCT ARGUMENTS Determine whetherthe following statement is sometimes, always, ornever true. Justify your reasoning.The volume of a cone with radius r and height hequals 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 theprism. 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 thebase area of the cone is 3 times as great as the basearea of the prism. For example, if the base of theprism has an area of 10 square units, then its volumeis 10h cubic units. So, the cone must have a base

area of 30 square units so that its volume is

or 10h cubic units.

ANSWER:

Sometimes; the statement is true if the base area ofthe cone is 3 times as great as the base area of theprism. For example, if the base of the prism has anarea of 10 square units, then its volume is 10h cubicunits. 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 arecalculating the volume of the cone below. Is either ofthem correct? Explain your answer.

SOLUTION:

The slant height is used for surface area, but theheight is used for volume. For this cone, the slantheight of 13 is provided, and we need to calculate theheight before we can calculate the volume. Alexandra incorrectly used the slant height.

ANSWER:

Cornelio; Alexandra incorrectly used the slant height.

38. CHALLENGE A cone has a volume of 568 cubiccentimeters. What is the volume of a cylinder thathas the same radius and height as the cone? Explainyour reasoning.

SOLUTION:

The volume of a cylinder would be 1704 cm3. Theformula for the volume of a cylinder is V = Bh, while

the formula for the volume of a cone is V = Bh. Thevolume of a cylinder is three times as much as thevolume of a cone with the same radius and height.

ANSWER:

1704 cm3; The volume of a cylinder is three times asmuch as the volume of a cone with the same radiusand height.

39. REASONING Give an example of a pyramid and aprism that have the same base and the same volume.Explain your reasoning.

SOLUTION:

The formula for volume of a prism is V = Bh and theformula for the volume of a pyramid is one-third ofthat. So, if a pyramid and prism have the same base,then in order to have the same volume, the height ofthe pyramid must be 3 times as great as the height ofthe prism.

Set the base areas of the prism and pyramid, andmake the height of the pyramid equal to 3 times theheight of the prism. Thus, a square pyramid with a base area of 16 and aheight of 12, a prism with a square base area of 16and a height of 4.

ANSWER:

Sample answer: A square pyramid with a base areaof 16 and a height of 12, a prism with a square basearea of 16 and a height of 4; if a pyramid and prismhave the same base, then in order to have the samevolume, the height of the pyramid must be 3 times asgreat as the height of the prism.



40. WRITING IN MATH Compare and contrastfinding volumes of pyramids and cones with findingvolumes of prisms and cylinders.

SOLUTION:

To find the volume of each solid, you must know thearea of the base and the height. The volume of apyramid is one third the volume of a prism that hasthe same height and base area. The volume of a coneis one third the volume of a cylinder that has thesame height and base area.

ANSWER:

To find the volume of each solid, you must know thearea of the base and the height. The volume of apyramid is one third the volume of a prism that hasthe same height and base area. The volume of a coneis one third the volume of a cylinder that has thesame height and base area.

41. Cullen is buying a tent that is in the shape of arectangular pyramid.

If the tent holds 88 cubic feet of air, how tall is thetent?

A

B

C D

SOLUTION: The volume of the tent is 88 cubic feet. Use thisinformation, along with the given dimensions, to findthe height of the tent by substituting these values intothe formula for the volume of a pyramid.

The correct choice is C.

ANSWER: C



42. Tara has a cylindrical candle mold that is 8 incheshigh with a diameter of 3 inches. She would like tomelt votive candles and reuse the wax. Each votivecandle is a cylinder with a height of 1.5 inches and aradius of 0.5 inch. How many votive candles areneeded to fill the candle mold?

A 12B 48C 57D 192E 226

SOLUTION: One way to begin is by finding the volume of thecylindrical candle mold and one votive candle.

Now, to determine how many votive candles neededto fill the cylindrical candle mold, divide the cylindricalvolume by the votive candle volume.

Tara will need the wax of 48 votive candles to fill thenew cylindrical candle mold. The correct choice is B.

ANSWER: B

43. A right circular cone has a height of 10 centimetersand a volume of 32 cubic centimeters. What is theradius of the cone?

A 0.5 cmB 1.75 cmC 2.25 cmD 2.75 cm

SOLUTION: Use the formula for the volume of a cone to find theradius.

The correct choice is B.

ANSWER: B



44. A conical sand toy has the dimensions shown. Howmany cubic centimeters of sand will it hold when it isfilled to the top?

A 12πB 15π

C

D

SOLUTION: First find the radius. The radius of the cone makes a right triangle with theheight and the slant height that is a PythagoreanTriple. The radius is 3 cm, the triangle is a 3-4-5triangle. Use the volume formula for a cone.

The correct choice is A.

ANSWER: A

45. MULTI-STEP The figure shows a cone with acylindrical hole cut out. What is the volume of thissolid?

SOLUTION: Use the volume formula for a cone and subtract thevolume of the cylinder

ANSWER:

891π cm3



46. What is the height of the pyramid if the volume is 210cubic feet and the base is 70 square feet?

SOLUTION: Use the formula for volume of a pyramid to find itsheight.

ANSWER: 9 ft



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