Holt CA Course 1 10-7Scaling Three-Dimensional Figures Warm Up Warm Up California Standards...

Holt CA Course 1

10-7 Scaling Three-Dimensional Figures

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Holt CA Course 1


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

Holt CA Course 1


MG2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.

California Standards

Holt CA Course 1


Holt CA Course 1


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 22 = 4 times as large as the 1 ft cube’s.

Holt CA Course 1


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

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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 larger cube is 9 times that of the smaller cube.

= 9

Holt CA Course 1



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 larger cube is 27 times that of the smaller cube.

= 27

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the edge lengths of the large and small cubes

Check It Out! Example 1A

2 cm cube1 cm cube

2 cm1 cm

Ratio of corresponding edges

The length of the edges of the larger cube is 2 times the length of the edges of the smaller cube.

= 2

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the surface areas of the two cubes

Check It Out! Example 1B

2 cm cube1 cm cube

24 cm2

6 cm2

Ratio of corresponding areas

The surface area of the larger cube is 4 times that of the smaller cube.

= 4

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the volumes of the two cubes

Check It Out! Example 1C

2 cm cube1 cm cube

8 cm3

1 cm3

Ratio of corresponding volumes

The volume of the larger cube is 8 times that of the smaller cube.

= 8

Holt CA Course 1


The surface area of a box is 1300 in2. What is the surface area of a similar box that is

smaller by a scale factor of ?

Additional Example 2A: Finding Surface Area and Volume of Similar Solids

12

S = 1300 · 12

2

= 1300 · 14

= 325 in2

Multiply by the square of the scale factor.

Simplify the power.

Multiply.

Holt CA Course 1


The volume of a child’s swimming pool is 28 ft3. What is the volume of a similar pool that is larger by a scale factor of 4?

Additional Example 2B: Finding Surface Area and Volume of Similar Solids

V = 28 · 43 Multiply by the cube of the scale factor.

= 28 · 64 Simplify the power.

= 1,792 ft3 Multiply.

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13

S = 1800 · 13

2

= 1,800 · 19

= 200

Multiply by the square of the scale factor.

Simplify the power.

Multiply.

Check It Out! Example 2A

The surface area of a box is 1800 in2. Find the surface area of a smaller, similarly shaped box

that has a scale factor of ?

Holt CA Course 1


Check It Out! Example 2B

The volume of a small hot tub is 48 ft3. What is the volume of a similar hot tub that is larger by a scale factor of 2?

V = 48 · 23 Use the volume of the smaller prismand the cube of the scale factor.

= 48 · 8 Simplify the power.

= 384 ft3 Multiply.

Holt CA Course 1


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.


30 8 = x

240 = x

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

Cross multiply.

Calculate the fill time.

30 s1 ft3

x 8 ft3

=

Holt CA Course 1


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?

Check It Out! Example 3


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

30 27 = x

810 = x

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

Cross multiply.

Calculate the fill time.

30 s1 ft3

x 27 ft3

= Set up a proportion and solve.

Holt CA Course 1



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

Holt CA Course 1


4. The surface area of a prism is 600 ft2. What is the surface area of a similar prism that is smaller by a scale factor of ?

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

66.7 ft2

60 min

1 3

Holt CA Course 1 10-7Scaling Three-Dimensional Figures Warm Up Warm Up California Standards...

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