Find The Volume Of Each Pyramid. - Brewton City

3Therefore, the volume of the cone is about 5.8 cm .ANSWER:12-5 Volumes of Pyramids and Cones35.8 cm23. 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.342,000,000 ft25. GARDENING The greenhouse is a regularoctagonal pyramid with a height of 5 feet. The basehas side lengths of 2 feet. What is the volume of thegreenhouse?SOLUTION:The volume of a circular cone is, wherer is the radius of the base and h is the height of thecone. Since the diameter of the cone is 8SOLUTION:The volume of a pyramid iscentimeters, the radius is or 4 centimeters. Theheight of the cone is 14 centimeters., where B is thearea 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 isor 45 , so the angle formed in the trianglebelow is 22.5 .3Therefore, the paper cone will hold about 234.6 cmof roasted peanuts.ANSWER:234.6 cm3Use a trigonometric ratio to find the apothem a.24. CCSS MODELING The Pyramid Arena inMemphis, Tennessee, is the third largest pyramid inthe world. It is approximately 350 feet tall, and itssquare base is 600 feet wide. Find the volume of thispyramid.The height of this pyramid is 5 feet.SOLUTION:The volume of a pyramid is, where B is thearea of the base and h is the height of the pyramid.Therefore, the volume of the greenhouse is about332.2 ft .ANSWER:42,000,000 ft325. GARDENING The greenhouse is a regularoctagonal pyramid with a height of 5 feet. The basehas side lengths of 2 feet. What is the volume of thegreenhouse?eSolutionsManual - Powered by CogneroANSWER:32.2 ft3Find the volume of each solid. Round to thenearest tenth.Page 10

Therefore, the volume of the greenhouse is about332.2 ft .ANSWER:12-5 Volumes of Pyramids and Cones332.2 ftANSWER:3190.6 m3Find the volume of each solid. Round to thenearest tenth.28.26.SOLUTION:SOLUTION:Volume of the solid given Volume of the smallcone Volume of the large coneANSWER:37698.5 cmANSWER:471.2 in329. 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?27.SOLUTION:SOLUTION:The building can be broken down into the rectangularbase and the pyramid ceiling. The volume of the baseisANSWER:The volume of the ceiling is3190.6 m3eSolutions Manual - Powered by CogneroPage 11The total volume is therefore 5000 1666.67 28.3SOLUTION:6666.67 ft . Two BTU's are needed for every cubic

6666.67 2 13,333 BTUs.ANSWER:12-5 Volumes of Pyramids and Cones37698.5 cmANSWER:13,333 BTUs29. 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?30. SCIENCE Marta is studying crystals that grow onrock formations.For a project, she is making a claymodel of a crystal with a shape that is a composite oftwo congruent rectangular pyramids. The base ofeach pyramid will be 1 by 1.5 inches, and the totalheight will be 4 inches.Determine the volume of the model. Explain whyknowing the volume is helpful in this situation.SOLUTION:The volume of a pyramid is, where B is thearea of the base and h is the height of the pyramid.SOLUTION:The building can be broken down into the rectangularbase and the pyramid ceiling. The volume of the baseisIt tells Marta how much clay is needed to make themodel.ANSWER:The volume of the ceiling is32 in ; It tells Marta how much clay is needed tomake the model.The total volume is therefore 5000 1666.67 36666.67 ft . 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 BTUs31. 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.30. SCIENCE Marta is studying crystals that grow onrock formations.For a project, she is making a claymodel of a crystal with a shape that is a composite oftwo congruent rectangular pyramids. The base ofeach pyramid will be 1 by 1.5 inches, and the totalheight will be 4 inches.Determine the volume of the model. Explain whyknowing the volume is helpful in this situation.a. Double h.SOLUTION:eSolutions Manual - Powered by CogneroThe volume of a pyramid is, where B is thearea of the base and h is the height of the pyramid.Page 12

ANSWER:a. The volume is doubled.2b. The volume is multiplied by 2 or 4.c. The volume is multiplied by 23 or 8.12-5 Volumes of Pyramids and Conesa. Double h.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. Findthe side length of the base.SOLUTION:The volume is doubled.The volume of a pyramid isb. Double r.area of the base and h is the height of the pyramid.Let s be the side length of the base., where B is the2The volume is multiplied by 2 or 4.c. Double r and h.The side length of the base is 15 cm.ANSWER:15 cm3volume is multiplied by 2 or 8.ANSWER:a. The volume is doubled.2b. The volume is multiplied by 2 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. Findthe side length of the base.33. The volume of a cone is 196π cubic inches and theheight 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 theheight of the cone, and r is the radius of the base.Since the diameter is 8 centimeters, the radius is 4centimeters.SOLUTION:The volume of a pyramid is, where B is thearea of the base and h is the height of the pyramid.Let s be the side length of the base.eSolutions Manual - Powered by CogneroPage 13The diameter is 2(7) or 14 inches.

The side length of the base is 15 cm.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.ANSWER:12-515Volumesof Pyramids and Conescm33. The volume of a cone is 196π cubic inches and theheight is 12 inches. What is the diameter?SOLUTION:The volume of a circular cone is, orSo, the radius is about 3.8 millimeters., where B is the area of the base, h is theheight of the cone, and r is the radius of the base.Since the diameter is 8 centimeters, the radius is 4centimeters.Use the Pythagorean Theorem to find the height ofthe cone.So, the height of the cone is about 4.64 millimeters.The volume of a circular cone isThe diameter is 2(7) or 14 inches., wherer is the radius of the base and h is the height of thecone.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:Therefore, the volume of the cone is about 70.23mm .ANSWER:370.2 mmThe 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.eSolutions Manual - Powered by CogneroSo, the radius is about 3.8 millimeters.35. MULTIPLE REPRESENTATIONS In thisproblem, you will investigate rectangular pyramids.a. GEOMETRIC Draw two pyramids withdifferent bases that have a height of 10 centimetersand a base area 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 thatmultiply to make 24.Sample answer:Page 14

c. ANALYTICAL Explain how multiplying the basearea and/or the height of the pyramid by 5 affects thevolume of the pyramid.12-5 Volumes of Pyramids and ConesSOLUTION:a. Use rectangular bases and pick values thatmultiply to make 24.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.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.ANSWER:a. Sample answer: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.eSolutions Manual - Powered by Cognerob. The volumes are the same. The volume of aPage 15pyramid 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 volumes

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 25.12-5 Volumes of Pyramids and Conesb. 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. CCSS ARGUMENTS Determine whether thefollowing statement is sometimes, always, or nevertrue. Justify your reasoning.The volume of a cone with radius r and height hequals the volume of a prism with height h.36. CCSS ARGUMENTS Determine whether thefollowing statement is sometimes, always, or nevertrue. 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 ofh iswhere B is the area of the base of theprism. Set the volumes equal.SOLUTION:The volume of a cone with a radius r and height h is. The volume of a prism with a height ofh iswhere B is the area of the base of theprism. Set the volumes equal.The volumes will only be equal when the radius ofthe cone is equal toor 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 basearea of 30 square units so that its volume isor 10h cubic units.The volumes will only be equal when the radius ofthe cone is equal toor 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 basearea of 30 square units so that its volume issquare units so that its volume isor 10hcubic units.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 cubiceSolutions Manual - Powered by Cognerounits. So, the cone must have a base area of 30square units so that its volume isANSWER: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 30or 10h37. ERROR ANALYSIS Alexandra and Cornelio arecalculating the volume of the cone below. Is either ofthem correct? Explain your answer.Page 16

area of 10 square units, then its volume is 10h cubicunits. So, the cone must have a base area of 30ANSWER:3square units so that its volume is12-5 Volumes of Pyramids and Conescubic units.or 10h37. ERROR ANALYSIS Alexandra and Cornelio arecalculating the volume of the cone below. Is either ofthem correct? Explain your answer.1704 cm ; The volume of a cylinder is three times asmuch as the volume of a cone with the same radiusand height.39. OPEN ENDED Give an example of a pyramid anda prism that have the same base and the samevolume. 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.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.Set the base areas of the prism and pyramid, andmake the height of the pyramid equal to 3 times theheight of the prism.Alexandra incorrectly used the slant height.ANSWER:Cornelio; Alexandra incorrectly used the slant height.38. REASONING 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:31704 cm ; The formula for the volume of a cylinderis V Bh, while the formula for the volume of a coneis V Bh. The volume of a cylinder is three timesas much as the volume of a cone with the sameradius and height.ANSWER:31704 cm ; The volume of a cylinder is three times asmuch as the volume of a cone with the same radiusand height.39. OPEN ENDED Give an example of a pyramid anda prism that have the same base and the samevolume. 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 ofeSolutions Manual - Powered by Cognerothe prism.Sample answer: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 acone is 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 acone is one third the volume of a cylinder that has thesame height and base area.Page 1741. A conical sand toy has the dimensions as shownbelow. How many cubic centimeters of sand will it

of 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 sametheofheightof the andpyramidmust be 3 times as12-5volume,VolumesPyramidsConesgreat 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 acone is 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 acone is one third the volume of a cylinder that has thesame height and base area.area 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 acone is one third the volume of a cylinder that has thesame height and base area.41. A conical sand toy has the dimensions as shownbelow. How many cubic centimeters of sand will ithold when it is filled to the top?A 12πB 15πCDSOLUTION:Use the Pythagorean Theorem to find the radius r ofthe cone.41. A conical sand toy has the dimensions as shownbelow. How many cubic centimeters of sand will ithold when it is filled to the top?A 12πB 15πCDSo, the radius of the cone is 3 centimeters.SOLUTION:Use the Pythagorean Theorem to find the radius r ofthe cone.The volume of a circular cone is, wherer is the radius of the base and h is the height of thecone.Therefore, the correct choice is A.ANSWER:AeSolutions Manual - Powered by CogneroSo, the radius of the cone is 3 centimeters.The volume of a circular cone is, where42. SHORT RESPONSE Brooke is buying a tent thatPageis18is in the shape of a rectangular pyramid. The base6 feet by 8 feet. If the tent holds 88 cubic feet of air,how tall is the tent’s center pole?

Therefore, the correct choice is A.12-5ANSWER:Volumes of Pyramids and ConesAANSWER:5.5 ft42. SHORT RESPONSE Brooke is buying a tent thatis in the shape of a rectangular pyramid. The base is6 feet by 8 feet. If the tent holds 88 cubic feet of air,how tall is the tent’s center pole?43. PROBABILITY A spinner has sections coloredred, blue, orange, and green. The table below

Find the volume of each pyramid. 62/87,21 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. 16:(5 75