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Geometry Volume of Prisms and Cylinders

Geometry Volume of Prisms and Cylinders. Warm Up. 1) Marcy, Rachel, and Tina went bowling. Marcy bowled 100 less than twice Rachel’s score. Tina bowled 40 more than Rachel’s score. Rachel bowled a higher score than Marcy. What is the greatest score that Tina could have bowled?

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Geometry Volume of Prisms and Cylinders

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  1. GeometryVolume of Prisms and Cylinders CONFIDENTIAL

  2. Warm Up 1) Marcy, Rachel, and Tina went bowling. Marcy bowled 100 less than twice Rachel’s score. Tina bowled 40 more than Rachel’s score. Rachel bowled a higher score than Marcy. What is the greatest score that Tina could have bowled? 2) Max can type 40 words per minute. He estimates that his term paper contains about 5000 words, and he takes a 15-minute break for every 45 minutes of typing. About how much time will it take Max to type his term paper? CONFIDENTIAL

  3. Volume of Prisms and Cylinders Thevolumeof a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. A cube built out of 27 unit cubes has a volume of 27 cubic units. Next Page: CONFIDENTIAL

  4. Cavalieri's principle says that if two three-dimensional figure have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume CONFIDENTIAL

  5. Volume of a Prism The volume of a prism with base area B and height h is V = Bh. B B h h Next Page: CONFIDENTIAL

  6. The volume of a cube with edge length s is V = s . The volume of a right rectangular prism with length l, width w, and height h is V = lwh. 3 s h w s l s CONFIDENTIAL

  7. Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. A). 8 cm 12 cm 10 cm volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh. V = lwh = (10)(12)(8) = 980 cm 3 Next Page: CONFIDENTIAL

  8. 8 cm 12 cm B). A cube with edge length 10 cm V= s Volume of a cube = 10 = 1000 cm Substitute 10 for s. 10 cm 3 3 3 Next Page: CONFIDENTIAL

  9. C). A right regular pentagonal Prism with base edge length 5 m and height 7 m. Find the apothem a of the base . First draw a right triangle on one base as shown. The measure of the angle with its vertex at the center is 360°/10 =36° Step:1 36° 7 m tan 36° = 2.5/a The leg of the triangle is half the side length. Or 2.5 m. a = 2.5/tan 36° Solve for a. 5 m Next Page: CONFIDENTIAL

  10. Use the value of a to find the base area. Step:2 36° Use the base area to find the volume. Step:3 7 m 5 m CONFIDENTIAL

  11. Now you try! 1) Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with legs 7 yd and 5 yd long. CONFIDENTIAL

  12. Marine Biology Application The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds. (Hint: 1 gallon = 0.134 ft ) 3 120 ft 8 ft 60 ft Next Page: CONFIDENTIAL

  13. Find the volume of the aquarium in cubic feet. Step:1 120 ft V = lwh = (120)(60)(8) = 57,600 cm 8 ft 3 60 ft Next Page: CONFIDENTIAL

  14. 120 ft Step:2 8 ft 60 ft Next Page: CONFIDENTIAL

  15. Step:3 120 ft 8 ft The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds 60 ft CONFIDENTIAL

  16. Now you try! 2) Estimate the volume in gallons and the weight of the water in the aquarium below if the height were doubled. 120 ft 8 ft 60 ft CONFIDENTIAL

  17. Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. CONFIDENTIAL

  18. The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r h. 2 h h r r CONFIDENTIAL

  19. Finding Volumes of Cylinders Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. A). 12 cm 8 cm Next Page: CONFIDENTIAL

  20. 2 A cylinder with a base area of 36 in and a height equal to twice the radius. B). Step:1 Use the base area to find the radius. Use the radius to find the height. The height is equal to twice the radius. Step:2 12 cm 8 cm Use the radius and height to find the volume. Step:3 CONFIDENTIAL

  21. Now you try! 3). Find the volume of a cylinder with a diameter of 16in. and a height of 17 in. Give your answer both in terms of and rounded to the nearest tenth. CONFIDENTIAL

  22. Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume. 6 m 12 m CONFIDENTIAL

  23. Now you try! 4) The length, width, and height of the prism are doubled. Describe the effect on the volume. 1.5 ft 3 ft 4 ft CONFIDENTIAL

  24. Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. 5 m 9 m 6 m 8 m CONFIDENTIAL

  25. Now you try! 5) Find the volume of the composite figure. Round to the nearest tenth. 3 cm 5 cm CONFIDENTIAL

  26. Now some problems for you to practice ! CONFIDENTIAL

  27. Assessment 1. Find the volume of each prism. B) A) 8 m 6 cm 4 cm 9 cm 6 m CONFIDENTIAL

  28. 2. The world’s largest ice cream cake, built in New York City on may 25, 2004, was approximately a 19 ft by 9 ft by 2 ft rectangular prism. Estimate the volume of the ice cream cake in gallons. If the density of the ice cream cake was 4.73 pounds per gallon, estimate the weight of the cake. (Hint: 1 gallon = 0.134 cubic feet) 19 ft 2 ft 9 ft CONFIDENTIAL

  29. 3. Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. B) A) 3 m 10 ft 5 m 12 ft CONFIDENTIAL

  30. 4. Describe the effect of each change on the volume of the given figure. A) The dimensions are multiplied by ¼ . B) The dimensions are tripled. 2 in. 8 ft 7 in. 4 ft 12 ft CONFIDENTIAL

  31. 5. Find the volume of each composite figure. Round to the nearest tenth. B) A) 10 in. 5 in. 6 ft 4 ft 15 in. 14 ft 4 ft 12 ft CONFIDENTIAL

  32. Let’s review CONFIDENTIAL

  33. Volume of Prisms and Cylinders Thevolumeof a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. A cube built out of 27 unit cubes has a volume of 27 cubic units. Next Page: CONFIDENTIAL

  34. Cavalieri's principle says that if two three-dimensional figure have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume CONFIDENTIAL

  35. Volume of a Prism The volume of a prism with base area B and height h is V = Bh. B B h h Next Page: CONFIDENTIAL

  36. The volume of a cube with edge length s is V = s . The volume of a right rectangular prism with length l, width w, and height h is V = lwh. 3 s h w s l s CONFIDENTIAL

  37. Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. A). 8 cm 12 cm 10 cm volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh. V = lwh = (10)(12)(8) = 980 cm 3 Next Page: CONFIDENTIAL

  38. 8 cm 12 cm B). A cube with edge length 10 cm V= s Volume of a cube = 10 = 1000 cm Substitute 10 for s. 10 cm 3 3 3 Next Page: CONFIDENTIAL

  39. C). A right regular pentagonal Prism with base edge length 5 m and height 7 m. Find the apothem a of the base . First draw a right triangle on one base as shown. The measure of the angle with its vertex at the center is 360°/10 =36° Step:1 36° 7 m tan 36° = 2.5/a The leg of the triangle is half the side length. Or 2.5 m. a = 2.5/tan 36° Solve for a. 5 m Next Page: CONFIDENTIAL

  40. Use the value of a to find the base area. Step:2 36° Use the base area to find the volume. Step:3 7 m 5 m CONFIDENTIAL

  41. Marine Biology Application The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds. (Hint: 1 gallon = 0.134 ft ) 3 120 ft 8 ft 60 ft Next Page: CONFIDENTIAL

  42. Find the volume of the aquarium in cubic feet. Step:1 120 ft V = lwh = (120)(60)(8) = 57,600 cm 8 ft 3 60 ft Next Page: CONFIDENTIAL

  43. 120 ft Step:2 8 ft 60 ft Next Page: CONFIDENTIAL

  44. Step:3 120 ft 8 ft The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds 60 ft CONFIDENTIAL

  45. Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. CONFIDENTIAL

  46. The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r h. 2 h h r r CONFIDENTIAL

  47. Finding Volumes of Cylinders Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. A). 12 cm 8 cm Next Page: CONFIDENTIAL

  48. 2 A cylinder with a base area of 36 in and a height equal to twice the radius. B). Step:1 Use the base area to find the radius. Use the radius to find the height. The height is equal to twice the radius. Step:2 12 cm 8 cm Use the radius and height to find the volume. Step:3 CONFIDENTIAL

  49. Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume. 6 m 12 m CONFIDENTIAL

  50. Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. 5 m 9 m 6 m 8 m CONFIDENTIAL

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