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Pre-Algebra 7-9 Scaling Three-Dimensional Figures 7-9 Scaling Three-Dimensional Figures Pre-Algebra Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures Warm Up Find the surface area of each rectangular prism. 1. length 14 cm, width 7 cm, height 7 cm 2. length 30 in., width 6 in., height 21 in 3. length 3 mm, width 6 mm, height 4 mm 4. length 37 in., width 9 in., height 18 in. 490 cm in mm 2 Pre-Algebra 7-9 Scaling Three-Dimensional Figures 2322 in 2

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures Problem of the Day A model of a solid-steel machine tool is built to a scale of 1 cm = 10 cm. The real object will weigh 2500 grams. How much does the model, also made of solid steel, weigh? 2.5 g

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures Learn to make scale models of solid figures.

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures Vocabulary capacity

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures Corresponding edge lengths of any two cubes are in proportion to each other because the cubes are similar. However, volumes and surface areas do not have the same scale factor as edge lengths. Each edge of the 2 ft cube is 2 times as long as each edge of the 1 ft cube. However, the cubes volume, or capacity, is 8 times as large, and its surface area is 4 times as large as the 1 ft cubes.

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures Multiplying the linear dimensions of a solid by n creates n 2 as much surface area and n 3 as much volume. Helpful Hint

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. A. the edge lengths of the large and small cubes Additional Example 1A: Scaling Models That Are Cubes 3 cm cube 1 cm cube 3 cm 1 cm Ratio of corresponding edges The edges of the large cube are 3 times as long as the edges of the small cube. = 3

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures B. the surface areas of the two cubes Additional Example 1B: Scaling Models That Are Cubes 3 cm cube 1 cm cube 54 cm 2 6 cm 2 Ratio of corresponding areas The surface area of the large cube is 9 times that of the small cube. = 9

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures C. the volumes of the two cubes Additional Example 1C: Scaling Models That Are Cubes 3 cm cube 1 cm cube 27 cm 3 1 cm 3 Ratio of corresponding volumes The volume of the large cube is 27 times that of the small cube. = 27

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures A 2 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. A. the edge lengths of the large and small cubes Try This: Example 1A 2 cm cube 1 cm cube 2 cm 1 cm Ratio of corresponding edges The edges of the large cube are 2 times as long as the edges of the small cube. = 2

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures B. the surface areas of the two cubes Try This: Example 1B 2 cm cube 1 cm cube 24 cm 2 6 cm 2 Ratio of corresponding areas The surface area of the large cube is 4 times that of the small cube. = 4

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures C. the volumes of the two cubes Try This: Example 1C 2 cm cube 1 cm cube 8 cm 3 1 cm 3 Ratio of corresponding volumes The volume of the large cube is 8 times that of the small cube. = 8

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures A box is in the shape of a rectangular prism. The box is 4 ft tall, and its base has a length of 3 ft and a width of 2 ft. For a 6 in. tall model of the box, find the following. A. What is the scale factor of the model? Additional Example 2: Scaling Models That Are Other Solid Figures The scale factor of the model is 1:8. Convert and simplify in. 4 ft = 6 in. 48 in. =

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures B. What are the length and the width of the model? Additional Example 2B: Scaling Models That Are Other Solid Figures Length: 3 ft = in. = 4 in Width: 2 ft = in. = 3 in The length of the model is 4 in., and the width is 3 in. 1 2

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures A box is in the shape of a rectangular prism. The box is 8 ft tall, and its base has a length of 6 ft and a width of 4 ft. For a 6 in. tall model of the box, find the following. A. What is the scale factor of the model? Try This: Example 2A The scale factor of the model is 1:16. Convert and simplify. 6 in. 8 ft = 6 in. 96 in. = 1 16

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures B. What are the length and the width of the model? Try This: Example 2B Length: 6 ft = in. = 4 in Width: 4 ft = in. = 3 in The length of the model is 4 in., and the width is 3 in. 1 2

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures It takes 30 seconds for a pump to fill a cubic container whose edge measures 1 ft. How long does it take for the pump to fill a cubic container whose edge measures 2 ft? Additional Example 3: Business Application V = 2 ft 2 ft 2 ft = 8 ft 3 Find the volume of the 2 ft cubic container. Set up a proportion and solve. Cancel units = x 240 = x It takes 240 seconds, or 4 minutes, to fill the larger container. Multiply. Calculate the fill time. 30 s 1 ft 3 x 8 ft 3 =

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures It takes 30 seconds for a pump to fill a cubic container whose edge measures 1 ft. How long does it take for the pump to fill a cubic container whose edge measures 3 ft? Try This: Example 3 Set up a proportion and solve. V = 3 ft 3 ft 3 ft = 27 ft 3 Find the volume of the 2 ft cubic container = x 810 = x It takes 810 seconds, or 13.5 minutes, to fill the larger container. Multiply. Calculate the fill time. 30 s 1 ft 3 x 27 ft 3 =

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures A 10 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. 1. the edge lengths of the two cubes 2. the surface areas of the two cubes 3. the volumes of the two cubes Lesson Quiz: Part 1

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Pre-Algebra 7-9 Scaling Three-Dimensional Figures 4. A pyramid has a square base measuring 185 m on each side and a height of 115 m. A model of it has a base 37 cm on each side. What is the height of the model? 5. A cement truck is pouring cement for a new 4 in. thick driveway. The driveway is 90 ft long and 20 ft wide. How long will it take the truck to pour the cement if it releases 10 ft 3 of cement per minute? Lesson Quiz: Part 2

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