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4. Contents Lesson 13-1Volumes of Prisms and Cylinders Lesson 13-2Volumes of Pyramids and Cones Lesson 13-3Volumes of Spheres Lesson 13-4Congruent and Similar Solids Lesson 13-5Coordinates in Space

5. Lesson 1 Contents Example 1Volume of a Triangular Prism Example 2Volume of a Rectangular Prism Example 3Volume of a Cylinder Example 4Volume of an Oblique Cylinder

6. Example 1-1a Find the volume of the triangular prism. V BhVolume of a prism 1500Simplify. Answer: The volume of the prism is 1500 cubic centimeters.

7. Example 1-1b Find the volume of the triangular prism. Answer:

8. Example 1-2a The weight of water is 0.029 pounds times the volume of water in cubic inches. How many pounds of water would fit into a rectangular child’s pool that is 12 inches deep, 3 feet wide, and 4 feet long? First, convert feet to inches.

9. Example 1-2b Answer: A rectangular child’s pool that is 12 inches deep, 3 feet wide, and 4 feet long, will hold about 601.3 pounds of water. To find the pounds of water that would fit into the child’s pool, find the volume of the pool. Now multiply the volume by 0.029. Simplify. V BhVolume of a prism 20,736The volume is 20,736 cubic inches. 36(48)(12)B 36(48), h 12

10. Example 1-2c The weight of water is 62.4 pounds per cubic foot. How many pounds of water would fit into a back yard pond that is rectangular prism 3 feet deep, 7 feet wide, and 12 feet long? Answer: 15,724.8 lb

11. Example 1-3a Find the volume of the cylinder to the nearest tenth. Answer: The volume is approximately 18.3 cubic centimeters. The height h is 1.8 centimeters, and the radius r is 1.8 centimeters. Use a calculator. Volume of a cylinder r 1.8, h 1.8

12. Example 1-3b Find the volume of the cylinder to the nearest tenth. The diameter of the base, the diagonal, and the lateral edge of the cylinder form a right triangle. Use the Pythagorean Theorem to find the height. Pythagorean Theorem Multiply. a h, b 8, and c 17

13. Example 1-3c Answer: The volume is approximately 754.0 cubic feet. Volume of a cylinder Now find the volume. Use a calculator. Subtract 64 from each side. Take the square root of each side. h 15 r 4 and h 15

14. Example 1-3d Answer: Find the volume of each cylinder to the nearest tenth. a. b.

15. Example 1-4a Find the volume of the oblique cylinder to the nearest tenth. Answer:The volume is approximately 17,671.5 cubic feet. Volume of a cylinder To find the volume, use the formula for a right cylinder. Use a calculator. r 15, h 25

16. Example 1-4b Find the volume of the oblique cylinder to the nearest tenth. Answer:

17. End of Lesson 1

18. Lesson 2 Contents Example 1Volume of a Pyramid Example 2Volumes of Cones Example 3Volume of an Oblique Cone

19. Example 2-1a Teofilo has a solid clock that is in the shape of a square pyramid. The clock has a base of 3 inches and a height of 7 inches. Find the volume of the clock. Volume of a pyramid Answer:The volume of the clock is 21 cubic inches. Multiply. 21 s 3, h 7

20. Example 2-1b Brad is building a model pyramid for a social studies project. The model is a square pyramid with a base edge of 8 feet and a height of 6.5 feet. Find the volume of the pyramid. Answer:

21. Example 2-2a Find the volume of the cone to the nearest tenth. Answer: The volume of the cone is approximately 314.2 cubic inches. Volume of a cone r = 5, h = 12 Use a calculator.

22. Example 2-2b Find the volume of the cone to the nearest tenth. Use trigonometry to find the radius of the base.

23. Example 2-2c Solve for r. Use a calculator. Now find the volume. Definition of tangent

24. Example 2-2d Answer: The volume of the cone is approximately 2167.6 cubic feet. Use a calculator. Volume of a cone

25. Example 2-2e Answer: Find the volume of each cone to the nearest tenth. a. b.

26. Example 2-3a Find the volume of the oblique cone to the nearest tenth.

27. Example 2-3b Answer:The volume of the oblique cone is approximately 192.1 cubic inches. Use a calculator. Volume of a cone r 4.2, h 10.4

28. Example 2-3c Find the volume of the oblique cone to the nearest tenth. Answer:

29. End of Lesson 2

30. Lesson 3 Contents Example 1Volumes of Spheres Example 2Volume of a Hemisphere Example 3Volume Comparison

31. Example 3-1a Find the volume of the sphere to the nearest tenth. Answer:The volume of the sphere is approximately 14,137.2 cubic centimeters. Volume of a sphere r = 15 Use a calculator.

32. Example 3-1b Find the volume of the sphere to the nearest tenth. First find the radius of the sphere. Circumference of a circle Solve for r. C 25

33. Example 3-1c Answer: The volume of the sphere is approximately 263.9 cubic centimeters. Volume of a sphere Use a calculator. Now find the volume.

34. Example 3-1d Answer: Find the volume of each sphere to the nearest tenth. a. b.

35. Example 3-2a Find the volume of the hemisphere. The volume of a hemisphere is one-half the volume of the sphere. Answer:The volume of the hemisphere is approximately 56.5 cubic feet. Volume of a hemisphere Use a calculator. r 3

36. Example 3-2b Find the volume of the hemisphere. Answer:

37. Example 3-3a Short-Response Test Item Compare the volumes of the sphere and the cylinder with the same radius and height as the radius of the sphere. Read the Test Item You are asked to compare the volumes of the sphere and the cylinder. r

38. Example 3-3b Answer:Since is greater than 1, the volume of the sphere is greater than the volume of the cylinder. Solve the Test Item h r Volume of the sphere: Volume of the cylinder: Simplify.

39. Example 3-3c Short-Response Test Item Compare the volumes of the hemisphere and the cylinder with the same radius and height as the radius of the hemisphere.

40. Example 3-3d Answer:The volume of the hemisphere is and the volume of the cylinder is Since is less than 1, the volume of the cylinder is greater than the volume of the hemisphere. The volume of the sphere is the volume of the cylinder.

41. End of Lesson 3

42. Lesson 4 Contents Example 1Similar and Congruent Solids Example 2Softballs and Baseballs

43. Example 4-1a Determine whether the pair of solids is similar, congruent, or neither.

44. Example 4-1a Find the ratios between the corresponding parts of the square pyramids. Simplify. Substitution

45. Example 4-1a Substitution Simplify. Answer:The ratios of the measures are equal, so we can conclude that the pyramids are similar. Since the scale factor is not 1, the solids are not congruent. Substitution

46. Example 4-1b Determine whether the pair of solids is similar, congruent, or neither.

47. Example 4-1b Compare the ratios between the corresponding parts of the cones. Simplify. Substitution Answer:Since the ratios are not the same, the cones are neither similar nor congruent.

48. Example 4-1c Determine whether each pair of solids is similar, congruent, or neither. a. Answer: similar

49. Example 4-1d b. Answer:neither

50. Example 4-2a Softballs and baseballs are both used to play a game with a bat. A softball has a diameter or 3.8 inches, while a baseball has a diameter of about 2.9 inches. Find the scale factor of the two balls. Write the ratio of the corresponding measures of the balls. Substitution Simplify. Answer:The scale factor is about 1.3:1.

51. Example 4-2b If the scale factor is a:b, then the ratio of the surface areas is Softballs and baseballs are both used to play a game with a bat. A softball has a diameter or 3.8 inches, while a baseball has a diameter of about 2.9 inches. Find the ratio of the surface areas of the two balls.

52. Example 4-2b Answer:The ratio of the surface areas is about 1.7:1. Simplify. Theorem 13.1

53. Example 4-2c If the scale factor is a : b, then the ratio of the volumes is Softballs and baseballs are both used to play a game with a bat. A softball has a diameter or 3.8 inches, while a baseball has a diameter of about 2.9 inches. Find the ratio of the volumes of the two balls.

54. Example 4-2c Answer:The ratio of the volumes of the two balls is about 2.2:1. Theorem 13.1 Simplify.

55. Example 4-2d Two sizes of balloons are being used for decorating at a party. When fully inflated, the balloons are spheres. The first balloon has a diameter of 18 inches while the second balloon has a radius of 7 inches.

56. a. Find the scale factor of the two balloons. b. Find the ratio of the surface areas of the two balloons. c. Find the ratio of the volumes of the two balloons. Example 4-2e Answer: 81:49 Answer: 729:343 Answer: 9:7

57. End of Lesson 4

58. Lesson 5 Contents Example 1Graph of a Rectangular Solid Example 2Distance and Midpoint Formulas in Space Example 3Translating a Solid Example 4Dilation with Matrices

59. Example 5-1a Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex. Plot the x-coordinate first. Draw a segment from the origin 3 units in the negative direction.

60. Example 5-1a Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex. To plot the y-coordinate, draw a segment 1 unit in the positive direction.

61. Example 5-1a Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex. Next, to plot the z-coordinate, draw a segment 2 units long in the positive direction.

62. Example 5-1a Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex. Label the coordinate A.

63. Example 5-1a Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex. Draw the rectangular prism and label each vertex. Answer:

64. Example 5-1b Graph the rectangular solid that contains the ordered triple N (1, 2, –3) and the origin. Label the coordinates of each vertex. Answer:

65. Example 5-2a Determine the distance between F(4, 0, 0) and G(–2, 3, –1). Answer: Distance Formula in Space Substitution Simplify.

66. Example 5-2b Determine the midpoint M of Midpoint Formula in Space Substitution Simplify. Answer:

67. Example 5-2b Answer: a. Determine the distance between A(0, –5, 0) and B(1, –2, –3). b. Determine the midpoint M of Answer:

68. Example 5-3a Suppose a two-story home is built with a bathroom on the first floor that is 9 feet wide, 6 feet deep, and 8 feet tall. Likewise, a bathroom on the second floor is directly above the one on the first floor and has the same dimensions. If the second floor is 10 feet above the first floor, find the coordinates of each vertex of the rectangular prism that represents the second floor bathroom. Explore Since the bathroom is a rectangular prism, use positive values for x, y, and z. Write the coordinates of each corner. The points of the bathroom will rise 10 feet for the points of the second-floor bathroom.

69. Example 5-3a Solve (6, 0, 10)D(6, 0, 0) (6, 9, 10)C(6, 9, 0) (0, 9, 10)B(0, 9, 0) (0, 0, 10)A(0, 0, 0) Translated coordinates, (x, y, z + 10) Image Coordinates of the vertices, (x, y, z) Preimage (0, 0, 18)E(0, 0, 8) (0, 9, 18)F(0, 9, 8) (6, 9, 18)G(6, 9, 8) (6, 0, 18)H(6, 0, 8) Plan Use a translation equation (x, y, z) (x, y, z + 10) to find the coordinates of each vertex of the rectangular prism that represents the second-floor bathroom.

70. Example 5-3a Answer:(0, 0, 10); (0, 9, 10); (6, 9, 10); (6, 0, 10); (0, 0, 18); (0, 9, 18); (6, 9, 18); (6, 0, 18) Examine Check that the distance between corresponding vertices is 10 feet.

71. Example 5-3b Suppose a warehouse has a room on the ground floor that is 20 feet wide, 25 feet long, and 12 feet tall. If the height of each floor is 12 feet, find the coordinates of each vertex of the rectangular prism that represents a room in the basement of the warehouse directly below the given room. Answer:(0, 0, –12); (0, 20, –12); (25, 20, –12); (25, 0, –12); (0, 0, 0); (0, 20, 0); (25, 20, 0); (25, 0, 0)

72. Example 5-4a First, write a vertex matrix for the rectangular prism. 22220000 44000440 20022200 HGFEDCBA z y x Dilate the prism by a scale factor of. Graph the image under the dilation.

73. Example 5-4a Next, multiply each element of the vertex matrix by the scale factor,. = 11110000 22000220 10011100 22220000 44000440 20022200 HGFEDCBA

74. Example 5-4a Answer: The coordinates of the vertices are A'(0, 0, 0), B'(0, 2, 0), C'(1, 2, 0), D'(1, 0, 0), E'(1, 0, 1), F'(0, 0, 1), G'(0, 2, 1), H'(1, 2, 1).

75. Example 5-4b Dilate the prism by a scale factor of 3. Graph the image under the dilation.

76. Example 5-4b Answer: The coordinates of the vertices are A'(0, 0, 0), B'(0, 3, 3), C'(0, 3, 0), D'(3, 3, 3), E'(0, 0, 3), F'(3, 0, 0), G'(3, 3, 0), H'(3, 0, 3).

77. End of Lesson 5

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