Pre-Algebra

7.9

Scaling Three-Dimensional Figures

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

Warm Up

Learn to make scale models of solid figures.

capacity

Vocabulary

Calculations

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.

Capacity

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

Helpful Hint

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 cube1 cm cube

3 cm1 cm

Ratio of corresponding edges

The edges of the large cube are 3 times as long as the edges of the small cube.

= 3

Example: Scaling Models That Are Cubes

B. the surface areas of the two cubes

3 cm cube1 cm cube

54 cm2

6 cm2

Ratio of corresponding areas

The surface area of the large cube is 9 times that of the small cube.

= 9

Example: Scaling Models That Are Cubes

C. the volumes of the two cubes

3 cm cube1 cm cube

27 cm3

1 cm3

Ratio of corresponding volumes

The volume of the large cube is 27 times that of the small cube.

= 27

Example: Scaling Models That Are Cubes

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 cube1 cm cube

2 cm1 cm

Ratio of corresponding edges

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

= 2

Try This

B. the surface areas of the two cubes

2 cm cube1 cm cube

24 cm2

6 cm2

Ratio of corresponding areas

The surface area of the large cube is 4 times that of the small cube.

= 4

Try This

C. the volumes of the two cubes

2 cm cube1 cm cube

8 cm3

1 cm3

Ratio of corresponding volumes

The volume of the large cube is 8 times that of the small cube.

= 8

Try This

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?

The scale factor of the model is 1:8.

Convert and simplify.18

6 in.4 ft

= 6 in.48 in.

=

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.18

36 8

12

Width: 2 ft = in. = 3 in.18

24 8

The length of the model is 4 in., and the width is 3 in.

12

Example: Scaling Models That Are Other Solid 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?

The scale factor of the model is 1:16.

Convert and simplify.6 in.8 ft

= 6 in.96 in.

= 116

Try This

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

Length: 6 ft = in. = 4 in. 116

7216

12

Width: 4 ft = in. = 3 in. 116

4816

The length of the model is 4 in., and the width is 3 in.

12

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 2 ft?

V = 2 ft 2 ft 2 ft = 8 ft3 Find the volume of the 2 ft cubic container.

Set up a proportion and solve.

Cancel units.

30 8 = x

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

Multiply.

Calculate the fill time.

30 s1 ft3

x 8 ft3

=

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 3 ft?

Set up a proportion and solve.

V = 3 ft 3 ft 3 ft = 27 ft3 Find the volume of the 2 ft cubic container.

30 27 = x

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

Multiply.

Calculate the fill time.

30 s1 ft3

x 27 ft3

=

Try This

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

100:1

10:1

1000:1

Lesson Quiz: Part 1

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

Lesson Quiz: Part 2