9.5 Volume of Pyramids

Transcription

1 9.5 Volume of Pyamids and Cones Goal Find the volumes of pyamids and cones. Key Wods pyamid p. 49 cone p. 49 volume p. 500 In the puzzle below, you can see that the squae pism can be made using thee conguent pyamids. The volume of each pyamid is onethid the volume of the pism. Volume Puzzle VOLUME OF A PYRAMID Wods Volume (aea of base)(height) h Symbols V h Stud t H lp STUDY TIP In Example (b), 6 m is the height of the tiangula base and 8 m is the height of the pyamid. EXAMPLE Find the Volume of a Pyamid Find the volume of the pyamid. a. b. 8 m 4 ft 7 m Solution a. V h Wite the fomula fo volume. b. V h 6 m (5 p 4)(6) Substitute. 2 p 7 p 6 (8) 40 Simplify. 56 ANSWER The volume is ANSWER The volume is 40 cubic feet. 56 cubic metes. 50 Chapte 9 Suface Aea and Volume

2 Find the Volume of a Pyamid Find the volume of the pyamid in. 0 cm 8 ft 9 ft 9 cm LOOK ACK Fo help with elating the volume of a pyamid to the volume of a pism, see p Volume of a Cone The volume of a cone is elated to the volume of a cylinde in the same way that the volume of a pyamid is elated to the volume of a pism. h h h h VOLUME OF A CONE Wods Volume (aea of base)(height) Symbols V h π 2 h h EXAMPLE 2 Find the Volume of a Cone Find the volume of the cone. Round you answe to the neaest whole numbe. Solution The adius of the cone is 8 cm. The height of the cone is h. 8 cm V π 2 h Wite the fomula fo volume of a cone. π(82 )(2) Substitute 8 fo and 2 fo h. 804 Multiply. ANSWER The volume is about 804 cubic centimetes. 9.5 Volume of Pyamids and Cones 5

3 EXAMPLE Find the Volume of a Cone What is the volume of the cone shown at the ight? in. 5 in. LOOK ACK Fo help with the Pythagoean Theoem, see p. 92. Solution You ae given the slant height of the cone. You need to find the height of the cone befoe you can find the volume. Find the height. (leg) 2 (leg) 2 (hypotenuse) 2 Use the Pythagoean Theoem. 2 h Substitute. 9 h 2 25 Simplify. h Subtact 9 fom each side. h 2 6 Simplify. h 6 Take the positive squae oot. h 4 Simplify. 2 Find the volume. V π 2 h Wite the fomula fo volume. π(2 )(4) Substitute fo and 4 fo h. 8 Multiply. ANSWER The volume is about 8 cubic inches. Find the Volume of a Cone Find the volume of the cone. Round you answe to the neaest whole numbe ft 9 in. 26 m 5 in. 7 ft 20 m 7. Find the volume of a cone with a height of 6 inches and a diamete of 8 inches. 8. Find the volume of a cone with a slant height of 7 feet and a diamete of 6 feet. 52 Chapte 9 Suface Aea and Volume

4 9.5 Execises Guided Pactice Vocabulay Check Match the solid with its volume fomula. Use each fomula once. A. Pyamid. Cone C. Pism D. Cylinde. V π 2 h 2. V h. V π 2 h 4. V h Skill Check Find the volume of the solid. If necessay, ound you answe to the neaest whole numbe in. 7 m ft m 5 m 9 in. 0 ft 0 ft 8. Find the volume of a cone with a slant height of 5 inches and a adius of 9 inches. Leave you answe in tems of π. Pactice and Applications Exta Pactice See p Find ase Aeas Find the aea of the base of the solid in. 4 in. 8 cm Volume of a Pyamid Find the volume of the pyamid in. 2 yd ft ft 8 in. 2 in. yd 9 yd Homewok Help Example : Exs. 2 6, Example 2: Exs. 7 9, 2, 26 Example : Exs. 0 2 Volume of a Pyamid Find the volume. 5. Find the volume of a pyamid with a base aea of 48 squae feet and a height of 5 feet. 6. Find the volume of a pyamid with a height of inches and a squae base with side lengths of 4 inches. 9.5 Volume of Pyamids and Cones 5

5 Volume of a Cone Find the volume of the cone. Round you answe to the neaest whole numbe cm 9. 0 yd 8 m yd 8 m Finding Volume Find the volume of the object. Round you answe to the neaest whole numbe cm 22. in. h 44 m 8 cm 2 46,55 m 2 Logical Reasoning In Execises 2 and 24, use a pyamid that has a height of 8 feet and a squae base with a side length of 6 feet. 2. How does the volume of the pyamid change if the base stays the same and the height is doubled? 24. How does the volume of the pyamid change if the height stays the same and the side length of the base is doubled? Eo Analysis In Execises 25 and 26, explain the student s eo and coect it in. 0 in. 0 in. V (0 p 0)() (00)(7) 4. in. 8 ft V π(8 2 )(6) π(64)(6) ft I CLASSZONE.COM HOMEWORK HELP Exta help with poblem solving in Exs is at classzone.com Finding Dimensions Find the missing dimension. 27. A pyamid has a volume of 20 cubic inches and the aea of the base is 5 squae inches. What is the height of the pyamid? 28. A cone has a volume of 8π cubic feet and a height of 6 feet. What is the adius of the base? 54 Chapte 9 Suface Aea and Volume 29. A pyamid with a squae base has a volume of 20 cubic metes and a height of 0 metes. What is a side length of the base?

6 Finding Volume with Slant Height Find the volume of the solid. If necessay, ound you answe to the neaest whole numbe m 8 ft 0 ft 4 in. 6 m 6 m 25 in. STUDY TIP Recall that the adius of the secto that foms a cone is the slant height of the cone. This is shown on p. 49. Using Nets In Execises 5, use the net to sketch the solid. Then find the volume of the solid. Round you answe to the neaest whole numbe m 5. 0 cm 6 cm 6 m Popcon A movie theate seves a small size of popcon in a conical containe and a lage size of popcon in a cylindical containe. 6. What is the volume of the small containe? What is the volume of in. the lage containe? in. 7. How many small containes of popcon do you have to buy to equal the amount of popcon in a lage containe? utte utte 8. Which containe gives you moe popcon fo you money? Explain you easoning. $2.00 $4.00 I CLASSZONE.COM HOMEWORK HELP Exta help with poblem solving in Exs. 9 4 is at classzone.com Pet Feede In Execises 9 4, use the diagam of the automatic pet feede. 9. Calculate the amount of food that can be placed in the feede. (Hint: Add the volume of the cylinde and the volume of the cone.) 40. If a cat eats cup of food each day, how much food does the cat eat in five days? Expess you answe in cubic inches. ( cup 4.4 in. ) 2.5 in. 7.5 in. 4 in. 4. Will the feede hold enough food fo the five days descibed in Execise 40? Explain. 9.5 Volume of Pyamids and Cones 55

7 Caee Volcanoes In Execises 42 44, use the infomation below..8 mi 0.25 mi 0.4 mi mi GEOLOGISTS collect and intepet data about volcanoes to help pedict when a volcano will eupt. Standadized Test Pactice efoe 980, Mount St. Helens was a cone-shaped volcano. In 980, Mount St. Helens eupted, destoying the cone-shaped tip. 42. What was the volume of Mount St. Helens befoe 980? 4. What was the volume of the cone-shaped tip that was destoyed? 44. What is the volume of Mount St. Helens today? (Hint: Subtact the volume of the tip fom the volume befoe 980.) 45. Multiple Choice A pyamid has a height of 9 yads and a volume of 96 cubic yads. Which of these ae possible dimensions fo its ectangula base? A 4 yads by 8 yads 2 yads by 8 yads C 6 yads by 7 yads D 5 yads by 8 yads Mixed Review Finding Cicumfeence and Aea Find the cicumfeence and the aea of the cicle. Round you answe to the neaest whole numbe. (Lesson 8.7) m 4 in. Suface Aea Find the suface aea of the figue. If necessay, ound you answe to the neaest whole numbe. (Lessons 9.2, 9.) in. 2 m 5 in. 8 in. 9 m Algeba Skills Slope Plot the points and daw a line though them. Then tell if the slope is positive, negative, zeo, o undefined. (Skills Review, p. 665) 52. (0, 0) and (4, ) 5. (5, 5) and (, 2) 54. (, 2) and (6, 2) 56 Chapte 9 Suface Aea and Volume 55. ( 2, 2) and (0, 4) 56. ( 4, ) and ( 4, 7) 57. (, 5) and (4, )