Scaling Three-Dimensional Figures9-9

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

Scaling Three-Dimensional Figures9-9

Warm UpFind 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

Scaling Three-Dimensional Figures9-9

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

Scaling Three-Dimensional Figures9-9

Prep for MA.8.G.5.1 …Convert units of measure between different measurement systems…and dimensions including…area…and derived units to solve problems.Rev MA.7.G.2.1

Sunshine State Standards

Scaling Three-Dimensional Figures9-9

Vocabulary

capacity

Scaling Three-Dimensional Figures9-9

Scaling Three-Dimensional Figures9-9

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 23 = 8 times as large, and its surface area is 4 times as large as the 1 ft cube’s.

Scaling Three-Dimensional Figures9-9

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

Helpful Hint

Scaling Three-Dimensional Figures9-9

A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values.

the edge lengths of the two cubes

Additional Example 1A: Scaling Models That Are Cubes

3 cm cube1 cm cube

3 cm1 cm

Ratio of corresponding edges

The length of the edges of the larger cube is 3 times the length of the edges of the smaller cube.

= 3

Scaling Three-Dimensional Figures9-9

A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values.

the surface areas of the two cubes

Additional Example 1B: Scaling Models That Are 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

Scaling Three-Dimensional Figures9-9

A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values.

the volumes of the two cubes

Additional Example 1C: Scaling Models That Are 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 smaller cube.

= 27

Scaling Three-Dimensional Figures9-9

An 8 cm cube is built from small cubes, each 2 cm on an edge. Compare the following values.

the edge lengths of the two cubes

Check It Out: Example 1A

8 cm cube2 cm cube

8 cm2 cm

Ratio of corresponding edges

The edges of the large cube are 4 times as long as the edges of the small cube.

= 4

Scaling Three-Dimensional Figures9-9

A 8 cm cube is built from small cubes, each 2 cm on an edge. Compare the following values.

the surface areas of the two cubes

Check It Out: Example 1B

8 cm cube2 cm cube

6(8 cm)2

6(2 cm)2

Ratio of corresponding areas

The surface area of the large cube is 42 = 16 times that of the small cube.

= 380 cm2

24 cm2 = 16

Scaling Three-Dimensional Figures9-9

A 8 cm cube is built from small cubes, each 2 cm on an edge. Compare the following values.

the volumes of the two cubes

Check It Out: Example 1C

Ratio of corresponding volumes

The volume of the large cube is 43 = 64 times that of the small cube.

8 cm cube2 cm cube

(8 cm)3

(2 cm)3

512 cm3

8 cm3 = 64

Scaling Three-Dimensional Figures9-9

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.

What is the scale factor of the model?

Additional Example 2A: Scaling Models That Are Other Solid Figures

The scale factor of the model is .

Convert and simplify.18

6 in.4 ft

= 6 in.48 in.

=

18

Scaling Three-Dimensional Figures9-9

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.

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.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

Scaling Three-Dimensional Figures9-9

A box is in the shape of a rectangular prism. The box is 5 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.

the scale of the model?

Check It Out: Example 2A

The scale of the model is 1:10.

6 in.5 ft

6 in.60 in.

= 110

Scaling Three-Dimensional Figures9-9

A box is in the shape of a rectangular prism. The box is 5 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.

the length and width of the model?

Check It Out: Example 2B

Length: 6 ft = in. 72 ft = 17 in. 110

110

12

Width: 4 ft = in. 48 ft = 4 in. 110

110

45

Scaling Three-Dimensional Figures9-9

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 ft3 Find the volume of the 2 ft cubic container.

Set up a proportion and solve.

Cancel units.

30 8 = x

240 = x

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

Multiply.

Calculate the fill time.

30 s1 ft3

x 8 ft3

=

Scaling Three-Dimensional Figures9-9

It takes 8 s for a machine to fill a cubic box whose edge measures 4 cm. How long would it take to fill a cubic box whose edge measures 10 cm?

Check It Out: Example 3

Vsmaller box = 4 cm 4 cm 4 cm = 64 cm3

It would take 125 seconds, or 2 minutes 5 seconds, to fill.

x s 1000 cm3

= ; 8000 = 64x, so x = = 125

Vlarger box = 10 cm 10 cm 10 cm = 1000 cm3

8 s 64 cm3

8000 64

Scaling Three-Dimensional Figures9-9

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Scaling Three-Dimensional Figures9-9

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 I

100:1

10:1

1000:1

Scaling Three-Dimensional Figures9-9

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?

Lesson Quiz: Part II

23 cm

60 min

Scaling Three-Dimensional Figures9-9

1. A 12 cm cube is built from small cubes, each 3 cm on an edge. Compare the edge lengths of the two cubes.

A. 12:1

B. 6:1

C. 4:1

D. 3:1

Lesson Quiz for Student Response Systems

Scaling Three-Dimensional Figures9-9

2. A 20 cm cube is built from small cubes, each 5 cm on an edge. Compare the surface areas of the two cubes.

A. 15:1

B. 16:1

C. 17:1

D. 18:1

Lesson Quiz for Student Response Systems

Scaling Three-Dimensional Figures9-9

3. The dimensions of a building are 140 m long, 125 m wide, and 200 m high. The scale model used to build the building is 14 cm long. What is the height of the model?

A. 12.5 cm

B. 20 cm

C. 125 cm

D. 200 cm

Lesson Quiz for Student Response Systems

Scaling Three-Dimensional Figures9-9

4. An aquarium has dimensions 5 ft long, 4 ft wide, and 6 ft deep. How long will it take to fill the aquarium with water from a pipe which releases 2 ft3 of water per minute?

A. 15 min

B. 30 min

C. 45 min

D. 60 min

Lesson Quiz for Student Response Systems