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4. Contents Lesson 11-1Three-Dimensional Figures Lesson 11-2Volume: Prisms and Cylinders Lesson 11-3Volume: Pyramids and Cones Lesson 11-4Surface Area: Prisms and Cylinders Lesson 11-5Surface Area: Pyramids and Cones Lesson 11-6Similar Solids Lesson 11-7Precision and Significant Digits

5. Lesson 1 Contents Example 1Identify Prisms and Pyramids Example 2Identify Diagonals and Skew Lines Example 3Analyze Real-World Drawings

6. Example 1-1a Identify the solid. Name the bases, faces, edges, and vertices. Answer: This figure has two parallel congruent bases that are rectangles, GHJK and LMNP, so it is a rectangular pyramid. faces: GHJK, LMNP, GHML, HJNM, JKPN, GKPL edges: vertices: G, H, J, K, L, M, N, P

7. Example 1-1b Identify the solid. Name the bases, faces, edges, and vertices. Answer: This figure has one triangular base, DEF, so it is a triangular pyramid. faces: DEF, DEG, DFG, EFG edges: vertices: D, E, F, G

8. Example 1-1c Identify each solid. Name the bases, faces, edges, and vertices. a. Answer: rectangular pyramid base: BCDE faces: ABC, ACD, ADE, AEB, BCDE edges: vertices: A, B, C, D, E

9. Example 1-1d Identify each solid. Name the bases, faces, edges, and vertices. b. Answer: rectangular prism bases: GHJK, LMNP or GKPL, HJNM or GHML, KJNP faces: GHJK, LMNP, GHML, HJNM, JKPN, GKPL edges: vertices: G, H, J, K, L, M, N, P

10. Example 1-2a Identify a diagonal and name all segments that are skew to it. Answer: is a diagonal because vertex Q and vertex W do not intersect any of the same faces;

11. Example 1-2b Identify a diagonal and name all segments that are skew to it. Answer:

12. Example 1-3a Architecture An architect’s sketch shows the plans for a new office building.

13. Example 1-3b Find the area of the ground floor if each unit on the drawing represents 55 feet. The drawing is 6  5, so the actual dimensions are 6(55)  5(55) or 330 feet by 275 feet. Formula for area Answer:The area of the ground floor is 90,750 square feet.

14. Example 1-3c How many floors are in the office building if each floor is 12 feet high? Assume each unit on the drawing represents 40 feet. You can see from the side view that the height of the building is 3 units. total height: number of floors: Answer:There are 10 floors in the office building.

15. Example 1-3d Architecture An architect’s sketch shows the plans for a new office building.

16. Example 1-3e a.Find the area of the ground floor if each unit represents 75 feet. b.How many floors are in the office building if each floor is 15 feet high? Assume each unit on the drawing represents 45 feet. Answer:168,750 ft 2 Answer:9 floors

17. End of Lesson 1

18. Lesson 2 Contents Example 1Volume of a Rectangular Prism Example 2Volume of a Triangular Prism Example 3Height of a Prism Example 4Volume of a Complex Solid Example 5Volume of a Cylinder

19. Example 2-1a Find the volume of the prism. Formula for volume of a prism The base is a rectangle, so Simplify. Answer:The volume is 3200 cubic centimeters.

20. Example 2-1b Find the volume of the prism. Answer:The volume is 45 ft 3.

21. Example 2-2a Find the volume of the triangular prism. Formula for volume of a prism The height of the prism is 3 in. B = area of base or.

22. Example 2-2b Simplify. Answer:The volume is 15 cubic inches.

23. Example 2-2c Find the volume of the triangular prism. Answer:The volume is 15 ft 3.

24. Example 2-3a Baking Cake batter is poured into a pan that is a rectangular prism whose base is an 8-inch square. If the cake batter occupies 192 cubic inches, what will be the height of the batter? Formula for volume of a prism Formula for volume of a rectangular prism Simplify. Divide each side by 64. Answer:The height of the batter is 3 inches.

25. Example 2-3b Swimming Pool A swimming pool is filled with 960 cubic feet of water. The pool is a rectangular prism 20 feet long and 12 feet wide and is the same depth throughout. Find the depth of the water. Answer:The water is 4 feet deep.

26. Example 2-4a Multiple-Choice Test Item Find the volume of the solid. A 262 m 3 B 918 m 3 C 972 m 3 D 1458 m 3 Read the Test Item The solid is made up of a rectangular prism and a triangular prism. The volume of the solid is the sum of both volumes.

27. Example 2-4b Solve the Test Item Step 1The volume of the rectangular prism is 12(9)(9) or 972 m 3. Step 2In the triangular prism, the area of the base is and the height is 12. Therefore, the volume is Step 3Add the volumes. Answer:The answer is D.

28. Example 2-4c Multiple-Choice Test Item Find the volume of the solid. A 932 in 3 B 896 in 3 C 1432 in 3 D 718 in 3 Answer:The answer is B.

29. Example 2-5a Find the volume of the cylinder. Round to the nearest tenth. Formula for volume of a cylinder Replace r with 7 and h with 14. Simplify. Answer:The volume is about 2155.1 cubic feet.

30. Example 2-5b Find the volume of the cylinder. Round to the nearest tenth. diameter of base 10 m, height 2 m Formula for volume of a cylinder Replace r with 5 and h with 2. Simplify. Answer:The volume is about 157.1 cubic meters. Since the diameter is 10 m, the radius is 5 m.

31. Example 2-5c Find the volume of each cylinder. Round to the nearest tenth. a. b.diameter of base 8 cm, height 6 cm Answer:351.9 in 3 Answer:301.6 cm 3

32. End of Lesson 2

33. Lesson 3 Contents Example 1Volumes of Pyramids Example 2Volume of a Cone Example 3Use Volume to Solve Problems

34. Example 3-1a Find the volume of the pyramid. If necessary, round to the nearest tenth. Formula for volume of a pyramid The base is a square, so The height of the pyramid is 12 inches. Simplify. Answer:The volume is 900 cubic inches.

35. Example 3-1b Find the volume of the pyramid. If necessary, round to the nearest tenth. base area 19 cm 2, height 21 cm Formula for volume of a pyramid Replace B with 19 and h with 21. Answer:The volume is 133 cubic centimeters. Simplify.

36. Example 3-1c Find the volume of each pyramid. If necessary, round to the nearest tenth. a. b.base area 32 cm 2, height 9 cm Answer:112 in 3 Answer:96 cm 3

37. Example 3-2a Find the volume of the cone. Round to the nearest tenth. Formula for volume of a cone Replace r with 5.5 and h with 8. Simplify. Answer:The volume is about 253.4 cubic meters.

38. Example 3-2b Find the volume of the cone. Round to the nearest tenth. Answer:422.7 in 3

39. Example 3-3a Landscaping When mulch was dumped from a truck, it formed a cone-shaped mound with a diameter of 15 feet and a height of 8 feet. What is the volume of the mulch? Formula for volume of a cone Since d = 15, replace r with 7.5. Replace h with 8. Answer:The volume of the mulch is about 471 cubic feet.

40. Example 3-3b Landscaping When mulch was dumped from a truck, it formed a cone-shaped mound with a diameter of 15 feet and a height of 8 feet. How many square feet can be covered with this mulch if 1 cubic foot covers 4 square feet of ground? Answer:1884 square feet can be covered with this mulch.

41. Example 3-3c Playground A load of wood chips for a playground was dumped and formed a cone-shaped mound with a diameter of 10 feet and a height of 6 feet. a.What is the volume of the wood chips? b.How many square feet of the playground can be covered with wood chips if 1 cubic foot of wood chips can cover 3 square feet of the playground? Answer:about 157 ft 3 Answer:471 ft 2

42. End of Lesson 3

43. Lesson 4 Contents Example 1Surface Area of a Rectangular Prism Example 2Surface Area of a Triangular Prism Example 3Surface Area of a Cylinder Example 4Compare Surface Areas

44. Example 4-1a Find the surface area of the rectangular prism. Write the formula. Substitution Simplify. Answer:The surface area of the rectangular prism is 1868 square centimeters.

45. Example 4-1b Find the surface area of the rectangular prism. Answer:444 in 2

46. Example 4-2a Find the surface area of the triangular prism. Find the area of each face. Bottom Left side Right side Two bases

47. Example 4-2b Add to find the total surface area. Answer:The surface area of the triangular prism is 336 square meters.

48. Example 4-2c Find the surface area of the triangular prism. Answer:96 ft 2

49. Example 4-3a Find the surface area of the cylinder. Round to the nearest tenth. Formula for surface area of a cylinder Replace r with 2.5 and h with 8. Simplify. Answer:The surface area of the cylinder is about 164.9 square meters.

50. Example 4-3b Find the surface area of the cylinder. Round to the nearest tenth. Answer:1504.4 in 2

51. Example 4-4a Cereals A company packages its cereal in a rectangular prism that is 2.5 inches by 7 inches by 12 inches. It is considering packaging it in a cylinder- shaped container having a 6-inch diameter and a height of 7.5 inches. Which uses the least amount of packaging? Surface area of rectangular prism top/bottomfront/backsides

52. Example 4-4b Surface area of cylinder top/bottomcurved surface Answer:Since 197.9 square inches < 263 square inches, the cylinder uses less packaging.

53. Example 4-4c Candy A candy company is deciding between two types of packaging for its gumballs. The first option is a rectangular prism that is 6 inches by 4 inches by 1.5 inches. The second option is a cylinder having a radius of 2 inches and a height of 5 inches. Which option requires less packaging? Answer:The rectangular prism requires less packaging. 78 < 88.0

54. End of Lesson 4

55. Lesson 5 Contents Example 1Surface Area of a Pyramid Example 2Use Surface Area to Solve a Problem Example 3Surface Area of a Cone

56. Example 5-1a Find the surface area of the square pyramid. Find the lateral area and the base area. Area of each lateral face Area of a triangle Replace b with 8 and h with 8.9. Simplify. There are 4 faces, so the lateral area is 4(35.6) or 142.4 square feet.

57. Example 5-1b Area of base Replace s with 8 and simplify. Area of a square The surface area of a pyramid equals the lateral area plus the area of the base. S142.464 Answer:The surface area of the square pyramid is 206.4 square feet.

58. Example 5-1c Find the surface area of the square pyramid. Answer:42 m 2

59. Example 5-2a Canopies A canopy is in the shape of a square pyramid that is 3.4 meters on each side. The slant height is 2 meters. How much canvas is used for the canopy? Find the lateral area only, since there is no bottom to the canopy. Area of each lateral face Formula for area of a triangle Replace b with 3.4 and h with 2.

60. Example 5-2b Simplify. One lateral face has an area of 3.4 square meters. There are 4 lateral faces, so the lateral area is 4(3.4) or 13.6 square meters. Answer:13.6 square meters of canvas was used to cover the canopy.

61. Example 5-2c Tent A tent is in the shape of a square pyramid that is 8 feet on each side. The slant height is 10 feet. Find the surface area of the tent. Answer:160 ft 2

62. Example 5-3a Find the surface area of the cone. Round to the nearest tenth. Formula for surface area of a cone Replace r with 3.5 and with 10. Simplify. Answer:The surface area of the cone is about 148.4 square feet.

63. Example 5-3b Find the surface area of the cone. Round to the nearest tenth. Answer:587.4 cm 2

64. End of Lesson 5

65. Lesson 6 Contents Example 1Identify Similar Solids Example 2Find Missing Measures Example 3Use Similar Solids to Solve a Problem

66. Example 6-1a Determine whether the pair of solids is similar. Write a proportion comparing radii and heights. Find the cross products. Simplify. Answer:The radii and heights are not proportional, so the cylinders are not similar.

67. Write a proportion comparing corresponding edge lengths. Find the cross products. Simplify. Answer:The corresponding measures are proportional, so the pyramids are similar. Example 6-1b Determine whether the pair of solids is similar.

68. Example 6-1c Determine whether the pair of solids is similar. a. Answer:yes

69. Example 6-1c Determine whether the pair of solids is similar. b. Answer:no

70. Example 6-2a The cylinders to the right are similar. Find the radius of cylinder A. Substitute the known values.

71. Example 6-2b Find the cross products. Simplify. Divide each side by 6. Answer:The radius of cylinder A is 6 centimeters.

72. Example 6-2c The rectangular prisms below are similar. Find the height of prism B. Answer:4.5 in.

73. Example 6-3a Doll Houses Lita made a model of her fish tank for her doll house. The model is exactlythe size of the original fish tank, whose dimensions are 120  30  38 cm. What is the volume of the model? ExploreYou know the scale factor and the volume of the fish tank is

74. Example 6-3b PlanSince the volumes have a ratio of, replace a with 1 and b with 25 in. Solve Write the ratio of volumes. Replace a with 1 and b with 25.

75. Example 6-3c Simplify. So, the volume of the tank is 15,625 times the volume of the model. Answer:The volume of the model is or about 8.8 cubic centimeters.

76. Example 6-3d ExamineCheck your answer by finding the dimensions of the model. Next, find the volume of the model using these dimensions.

77. Example 6-3e Trains A scale model of a railroad boxcar is in the shape of a rectangular prism and is the size of the actual boxcar. The scale model has a volume of 72 cubic inches. What is the volume of the actual boxcar? Answer:9,000,000 in 3

78. End of Lesson 6

79. Lesson 7 Contents Example 1Identify Precision Units Example 2Identify Significant Digits Example 3Add Measurements Example 4Multiply Measurements

80. Example 7-1a Identify the precision unit of the thermometer shown on the right. Answer:The precision unit is 5°F.

81. Example 7-1b Identify the precision unit of the ruler shown on the right. Answer:

82. Example 7-2a Determine the number of significant digits in 1040 miles. Answer:3 significant digits

83. Example 7-2b Determine the number of significant digits in 0.003 centimeter. Answer:1 significant digit

84. Example 7-2c Determine the number of significant digits in 90.051 kilograms. Answer:5 significant digits

85. Example 7-2d Determine the number of significant digits in 0.06300 liter. Answer:4 significant digits

86. Determine the number of significant digits in each measure. a. 34.70 inches b.0.000003 mile c.2300 centimeters d.2.08 meters Example 7-2e Answer:4 Answer:1 Answer:2 Answer:3

87. Example 7-3a The sides of a quadrilateral measure 0.6 meter, 0.044 meter, 0.024 meter, and 0.103 meter. Use the correct number of significant digits to find the perimeter. 0.6  1 decimal place 0.044  3 decimal places 0.024  3 decimal places +0.103  3 decimal places 0.771 The least precise measurement, 0.6, has one decimal place. So, round 0.771 to one decimal place, 0.8. Answer:The perimeter of the quadrilateral is about 0.8 meter.

88. Example 7-3b The sides of a triangle measure 2.04 centimeters, 3.2 centimeters, and 2.625 centimeters. Use the correct number of significant digits to find the perimeter. Answer:7.9 cm

89. Example 7-4a What is the area of the bedroom shown here? 12.25  4 significant digits 171.5  4 significant digits x 14  2 significant digits To find the area, multiply the length and the width.

90. Example 7-4a The answer cannot have more significant digits than the measurements of the length and width. So, round 171.5 square feet to 2 significant digits. Answer:The area of the bedroom is about 170 square feet.

91. Example 7-4b Suppose a bedroom was 13.75 feet wide and 12.5 feet long. What would be the area of the bedroom? Answer:171 ft 2

92. End of Lesson 7

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