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**Scaling Three-Dimensional Figures**

7.9 Pre-Algebra

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**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 cm2 1872 in2 108 mm2 2322 in2

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**Learn to make scale models of solid figures.**

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Vocabulary capacity

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Calculations

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Capacity 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 cube’s volume, or capacity, is 8 times as large, and its surface area is 4 times as large as the 1 ft cube’s.

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Multiplying the linear dimensions of a solid by n creates n2 as much surface area and n3 as much volume. Helpful Hint

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**Example: Scaling Models That Are Cubes**

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 3 cm cube 1 cm cube 3 cm 1 cm Ratio of corresponding edges = 3 The edges of the large cube are 3 times as long as the edges of the small cube.

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**Example: Scaling Models That Are Cubes**

B. the surface areas of the two cubes 3 cm cube 1 cm cube 54 cm2 6 cm2 Ratio of corresponding areas = 9 The surface area of the large cube is 9 times that of the small cube.

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**Example: Scaling Models That Are Cubes**

C. the volumes of the two cubes 3 cm cube 1 cm cube 27 cm3 1 cm3 Ratio of corresponding volumes = 27 The volume of the large cube is 27 times that of the small cube.

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Try This 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 2 cm cube 1 cm cube 2 cm 1 cm Ratio of corresponding edges = 2 The edges of the large cube are 2 times as long as the edges of the small cube.

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**Try This B. the surface areas of the two cubes 2 cm cube 1 cm cube**

Ratio of corresponding areas = 4 The surface area of the large cube is 4 times that of the small cube.

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**Try This C. the volumes of the two cubes 2 cm cube 1 cm cube 8 cm3**

Ratio of corresponding volumes = 8 The volume of the large cube is 8 times that of the small cube.

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**Example: Scaling Models That Are Other Solid 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? 6 in. 4 ft = 6 in. 48 in. = 1 8 Convert and simplify. The scale factor of the model is 1:8.

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**Example: Scaling Models That Are Other Solid Figures**

B. What are the length and the width of the model? Length: 3 ft = in. = 4 in. 1 8 36 2 Width: 2 ft = in. = 3 in. 1 8 24 The length of the model is 4 in., and the width is 3 in. 1 2

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Try This 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? 6 in. 8 ft = 6 in. 96 in. = 1 16 Convert and simplify. The scale factor of the model is 1:16.

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**Try This B. What are the length and the width of the model?**

Length: 6 ft = in. = 4 in. 1 16 72 2 Width: 4 ft = in. = 3 in. 1 16 48 The length of the model is 4 in., and the width is 3 in. 1 2

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**Example: Business Application**

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? Find the volume of the 2 ft cubic container. V = 2 ft 2 ft 2 ft = 8 ft3 Set up a proportion and solve. 30 s 1 ft3 x 8 ft3 = Cancel units. 30 8 = x Multiply. 240 = x Calculate the fill time. It takes 240 seconds, or 4 minutes, to fill the larger container.

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Try This 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? Find the volume of the 2 ft cubic container. V = 3 ft 3 ft 3 ft = 27 ft3 Set up a proportion and solve. 30 s 1 ft3 x 27 ft3 = 30 27 = x Multiply. 810 = x Calculate the fill time. It takes 810 seconds, or 13.5 minutes, to fill the larger container.

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**Lesson Quiz: Part 1 1. the edge lengths of the two cubes**

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 10:1 100:1 1000:1

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Lesson Quiz: Part 2 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 ft3 of cement per minute? 23 cm 60 min

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