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Similar SolidsA

Multi-Part Lesson

9-2PART B C

528 Chapter 9 Measurement and Proportional Reasoning

Changes in DimensionsMODELS Stephen is creating a model of the Washington Monument for history class. The model will be 1 _

100 of the monument’s

actual size.

1. The pyramid that sits atop the monument’s obelisk shape has a height of 55.5 feet. What is the height of the pyramid on the model Stephen is creating? 0.555 ft

2. MAKE A CONJECTURE Write a sentence about the area of the triangular side of the model compared with the actual monument. ( 1 _

100 )

2 or 1 _

10,000 of the actual area

Cubes are similar solids because they have the same shape and their corresponding linear measures are proportional.

The cubes at the right are similar.

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4 in. 8 in.

8 in.

8 in.

4 in.

4 in.

The ratio of their corresponding

edge lengths is 8 _ 4 or 2. The scale

factor is 2. How are their surface areas related?

S.A. of Small Cube S.A. of Large CubeS.A. = 6(4)(4)

There are 6 faces.

S.A. = 6(2 · 4)(2 · 4) = 2 · 2(6)(4 · 4) = 22(6)(4 · 4)

To find the surface area of the large cube, multiply the surface area of the small cube by the square of the scale factor, 22 or 4. This relationship is true for any similar solids.

Surface Area of Similar Solids

If Solid X is similar to Solid Y by a scale factor, then the surface area of X is equal to the surface area of Y times the square of the scale factor.

Main IdeaSolve problems involving similar solids.

New Vocabularysimilar solids

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Lesson 9-2 Similar Solids 529

Surface Area of Similar Solids

The surface area of a rectangular prism is 78 square centimeters.

What is the surface area of a similar prism that is 3 times as large?

S.A. = 78 × 32 Multiply by the square of the scale factor.

S.A. = 78 × 9 Square 3.

S.A. = 702 cm2 Simplify.

MODELS Refer to page 528. The surface area of the exposed portion

of the pyramid atop the Washington Monument is 4,012 square feet. What is the surface area in square inches, to the nearest tenth, of the pyramid on Stephen’s model?

S.A. = 4,012 × ( 1 _ 100

) 2 Multiply by the square of the scale factor.

S.A. = 4,012 × 1 _ 10,000

Square 1 _ 100 .

S.A. = 0.4012 ft2 Simplify.

S.A. = 0.4012 ft · ft × 12 in. _ 1 ft

× 12 in. _ 1 ft

Convert to inches.

S.A. = 57.8 in2 Simplify.

The surface area of Stephen’s model is 57.8 square inches.

a. RAISINS The world’s largest box of raisins is located in

Kingsburg, California. Its surface area is 352 square feet. If a similar box is smaller than the Kingsburg box by a scale factor

of 1 _ 48

, what is its surface area? 11 _ 72

ft2 or 0.15 ft2

The volumes of similar solids are also related. Refer to the cubes on page 528.

Volume of Small Cube Volume of Large CubeV = 4 · 4 · 4 V = (2 · 4)(2 · 4)(2 · 4)

= 2 · 2 · 2(4 · 4 · 4) = 23(4 · 4 · 4)

The volumes of similar solids are related by the cube of the scale factor.

Volume of Similar Solids

If Solid X is similar to Solid Y by a scale factor, then the volume of X is equal to the volume of Y times the cube of the scale factor.

Real-World LinkIt takes about 4 1 _ 2 pounds of fresh grapes to make one pound of raisins.

Real-World Link

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530 Chapter 9 Measurement and Proportional Reasoning

Volume of Similar Solids

A triangular prism has a volume of 432 cubic yards. If the prism

is reduced to one third its original size, what is the volume of the new prism?

V = 432 × ( 1 _ 3 )

3 Multiply by the cube of the scale factor.

V = 432 × 1 _ 27

Cube 1 _ 3 .

V = 16 yd3 Simplify.

The volume of the new prism is 16 cubic yards.

b. A square pyramid has a volume of 512 cubic centimeters. What

is the volume of a square pyramid with dimensions one fourth of the original? 8 cm3

HOCKEY The standard hockey puck

measures as shown at the right. Find

1 in.

1.5 in.

the surface area and volume of the giant puck at the left.

Find the volume and surface area of the standard puck first.

V = πr2h S.A. = 2(πr2) + 2πrh

≈ (3.14)(1.5)2(1) ≈ 2(3.14)(1.5)2 + 2(3.14)(1.5)(1)

≈ 7.065 in3 ≈ 14.13 + 9.42

≈ 23.55 in2

Find the volume and surface area of the giant puck using the scale factor.

V = V(40)3 S.A. = S.A.(40)2

= (7.065)(40)3 = (23.55)(40)2

= 452,160 in3 = 37,680 in2

The giant hockey puck has a volume of 452,160 cubic inches and a surface area of 37,680 square inches.

c. The dimensions of a rectangular solid are 10 inches by 7 inches

by 6 inches. Find the surface area and volume of a solid that is larger by a scale factor of 25 and similar to the original solid.

Real-World LinkThe hockey puck that appears to be crashing into the side of the wall at Nationwide Arenain Columbus, Ohio, is about 40 timesthe actual size of a standard puck.

Real-World LinkThe hockey puck that

c. 215,000 in2; 6,562,500 in3

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Lesson 9-2 Similar Solids 531

Example 1(p. 529)

1. The surface area of a rectangular prism is 35 square inches. What is the surface area of a similar solid that has been enlarged by a scale factor of 7? 1,715 in2

Example 2(p. 529)

2. MODELS The surface area of a ship’s hull is about 11,000 square meters. What is the surface area, to the nearest tenth, of the hull of a model ship that is smaller by a scale factor of 1_

100 ? 1.1 m2

Example 3(p. 530)

3. The volume of a cylinder is about 425 cubic centimeters. What is the volume, to the nearest tenth, of a similar solid that is smaller by a scale

factor of 1_3

? 15.7 cm3

Example 4(p. 530)

4. ART STUDIO The sink in Josh’s art studio measures 16 inches by 15 inches by 6 inches. A second sink used just for paint brushes has a similar shape and is smaller by a scale factor of 1_

2 . Find the surface area and volume of

the second sink. 213 in2; 180 in3

= Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP2.

Example 1(p. 529)

5. The surface area of a rectangular prism is 1,300 square inches. Find the surface area of a similar solid that is larger by a scale factor of 3. 11,700 in2

6. The surface area of a triangular prism is 10.4 square meters. What is the surface area of a similar solid that is smaller by a scale factor of 1_

4 ? 0.65 m2

Example 2(p. 529)

7 FOOD A cereal box has a surface area of 280 square inches. What is the surface area of a similar box that is larger by a scale factor of 1.4? 548.8 in2

8. DISPLAYS A glass display box has a surface area of 378 square inches. How many square inches of glass are used to create a glass display box with dimensions one-half the original? 94.5 in2

Example 3(p. 530)

9. A cone has a volume of 9,728 cubic millimeters. What is the volume of a similar cone one-eighth the size of the original? 19 mm3

10. A triangular prism has a volume of 350 cubic meters. If the dimensions are tripled, what is the volume of the new prism? 9,450 m3

Example 4(p. 530)

11. ARCHITECTURE The model of a new

18 in.

10 in.

28 in.

apartment building is shown. The architect plans for the building to be 144 times the size of the model. What will be the surface area and volume of the new building when it is completed? 251,712 ft2; 8,709,120 ft3

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532 Chapter 9 Measurement and Proportional Reasoning

12. The world’s largest cube puzzle is in Knoxville, Tennessee. It measures 6 feet on each side and is 24 times the size of an actual puzzle. Find the surface area and volume of the standard cube puzzle. See margin.

13 Two spheres are similar in shape. The scale factor between the smaller

B sphere and the larger sphere is 3 _ 4 . If the volume of the smaller sphere is

126.9 cubic meters, what is the volume of the larger sphere? 300.8 m3

Determine whether each statement is always, sometimes, or never true.

14. Two prisms with equal bases are similar. sometimes

15. Similar solids have equal volumes. sometimes

16. Two cubes are similar. always

17. A prism and pyramid are similar. never

18. Find the missing

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10.5 ft 3 ft

7 ft1.5 ft

y

x

measure for the pair of similar solids. x = 2, y = 5.25

19. Two similar cylinders

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18 cm

6 cm

Cylinder A Cylinder B

V = 1130.4 cm3S.A. = 5,425.92 cm2

are shown.

a. What is the ratio of their radii? 3 : 1

b. What is the ratio of their surface areas and their volumes?

c. Find the surface area of Cylinder B. 602.88 cm2

d. Find the volume of Cylinder A. 30,520.8 cm3

20. GRAPHIC NOVEL Refer to the graphic novel frame below. Use the distances

and times you converted in Lesson 9-1D and the formula r = d _ t to

calculate the different speeds and express them in miles per hour.

Now we are finding how fast the cars were traveling. You will need the distances and times you converted in Lesson 9-1D.

19b. surface area, 9 : 1; volume, 27 : 119b. surface area, 9 : 1; volume, 27 : 1

Real-World LinkCube Puzzles The world’s largest automated cube puzzle solves itself every 30 seconds. The record time a standard cube puzzle was solved in 2008 was 7.08 seconds.

14.Real-World Link

14.Cube Puzzles 14.

12.

20. 4.37 mi/h; 5.68 mi/h; 6.81 mi/h; 8.42 mi/h

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Lesson 9-2 Similar Solids 533

C 21. CHALLENGE A frustum is the solid left

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frustum1.5 in.

3 in.

3 in.

6 in.after a cone is cut by a plane parallel to its base and the top cone is removed.

a. Is the smaller cone that is removed similar to the original cone? Justify your response.

b. What is the volume of the smaller cone? the larger cone? 7.065 in3; 56.52 in3

c. What is the ratio of the volume of the smaller cone to the volume of the larger cone? 1:8

d. What is the volume of the frustum? 49.455 in3

22. Explain what happens to the volume of a prism when its dimensions are tripled. The volume is 27 times greater.

25. ART Julianna is making a clay figurine of a dog. The dog is 75 centimeters tall. If she uses a scale of 1 centimeter = 10 centimeters, how tall will the clay figurine be? (Lesson 9-2A) 7.5 cm

26. SPORTS The table shows the dimensions of the fields used Sport Length

(yards)Width(yards)

Field hockey 60 100

Football 53 1 _ 3 120

Lacrosse 60 110

Soccer 70 115

in various sports. (Lesson 9-1F) a. 54,000 ft2 b. 13,050 ft2

a. What is the area of the field hockey field in square feet?

b. What is the difference between the area of the soccer field and the area of the lacrosse field in square feet?

c. If an acre is 43,560 square feet, about how many acres are all four fields combined? about 5.6 acres

Practice

23. For the similar pyramids, find the ratio of the surface area of the larger pyramid to the smaller pyramid. C

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25 cm 15 cm

A. 5 _ 3 C. 25 _

9

B. 25 _ 15

D. 10 _ 6

24. Two similar prisms have volumes of4 cubic meters and 864 cubic meters, respectively. How many times larger is the second prism? F

F. 6 times larger

G. 16 times larger

H. 96 times larger

I. 216 times larger

21a. Yes, the ratios 3 _ 6

and 1.5 _ 3 are equal.

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