7-9 Scaling Three-Dimensional Figures

Learn to make scale models of solid figures.

Pre-Algebra

7-9 Scaling Three-Dimensional Figures

Pre-Algebra

7-9 Scaling Three-Dimensional Figures

Helpful Hint Multiplying the linear dimensions of a solid by n creates n 2 as much surface area and n 3 as much volume.

Pre-Algebra

7-9 Scaling Three-Dimensional Figures Example 1A: 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 = 3 Ratio of corresponding edges The edges of the large cube are 3 times as long as the edges of the small cube.

Pre-Algebra

7-9 Scaling Three-Dimensional Figures Example 1B: Scaling Models That Are Cubes

B. the surface areas of the two cubes 3 cm cube 1 cm cube 54 cm 6 cm 2 2 = 9 Ratio of corresponding areas The surface area of the large cube is 9 times that of the small cube.

Pre-Algebra

7-9 Scaling Three-Dimensional Figures Example 1C: Scaling Models That Are Cubes

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

Pre-Algebra

7-9 Scaling Three-Dimensional Figures

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 cube’s volume, or 1 ft cube’s.

capacity , is 8 times as large, and its surface area is 4 times as large as the Pre-Algebra

7-9 Scaling Three-Dimensional Figures Example 2: 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.

= 1 8 Convert and simplify.

The scale factor of the model is 1:8.

Pre-Algebra

7-9 Scaling Three-Dimensional Figures Example 2B: Scaling Models That Are Other Solid Figures

B. What are the length and the width of the model?

Length:  3 ft Width:  2 ft The length of the model is 4 in., and the width is 3 in.

Pre-Algebra

7-9 Scaling Three-Dimensional Figures Example 3: 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?

V = 2 ft  2 ft  2 ft = 8 ft 3 Find the volume of the 2 ft 30 s 1 ft 3 = x 8 ft 3 30  8 = x Cancel units.

Multiply.

240 = It takes 240 seconds, or 4 minutes, to fill the larger container.

x Calculate the fill time.

Pre-Algebra

7-9 Scaling Three-Dimensional Figures Try This: Example 3

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?

V = 3 ft  3 ft  3 ft = 27 ft 3 Find the volume of the 2 ft Set up a proportion and solve.

cubic container.

30  30 s 1 ft 3 = x 27 ft 3 27 = x Multiply.

810 = It takes 810 seconds, or 13.5 minutes, to fill the larger container.

x Calculate the fill time.

Pre-Algebra

7-9 Scaling Three-Dimensional Figures

Pre-Algebra

7-9 Scaling Three-Dimensional Figures Try This: Example 1A

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

Pre-Algebra

7-9 Scaling Three-Dimensional Figures Try This: Example 1B

B. the surface areas of the two cubes 2 cm cube 1 cm cube 24 cm 6 cm 2 2 = 4 Ratio of corresponding areas The surface area of the large cube is 4 times that of the small cube.

Pre-Algebra

7-9 Scaling Three-Dimensional Figures Try This: Example 1C

C. the volumes of the two cubes 2 cm cube 1 cm cube 8 cm 1 cm 3 3 = 8 Ratio of corresponding volumes The volume of the large cube is 8 times that of the small cube.

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 cube’s volume, or 1 ft cube’s.

capacity , is 8 times as large, and its surface area is 4 times as large as the Pre-Algebra