Multi-Part Lesson

2

PART

Similar Solids A

Main Idea Solve problems involving similar solids.

Vocabulary V ssimilar solids

Get ConnectED

B

C

Changes in Dimensions MODELS Stephen is creating a model of the W Washington Monument for history class. 1 The model will be _ of the monument’s 100 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? 2. MAKE A CONJECTURE Write a sentence about the area of the triangular side of the model compared with the actual monument. Cubes are similar solids because they have the same shape and their corresponding linear measures are proportional. The cubes at the right are similar. The ratio of their corresponding

8 in.

8 edge lengths is _ or 2. The scale 4

factor is 2. How are their surface areas related? S.A. of Small Cube S.A. = 6(4)(4) There are 6 faces.

4 in. 4 in.

4 in.

8 in. 8 in.

S.A. of Large Cube 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.

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Measurement and Proportional Reasoning

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 the previous page. 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?

( 100 )2

1 S.A. = 4,012 × _

Multiply by the square of the scale factor.

1 S.A. = 4,012 × _

Square _.

S.A. = 0.4012 ft2

Simplify.

12 in. 12 in. S.A. = 0.4012 ft · ft × _ ×_ 1 ft 1 ft

Convert to inches.

S.A. = 57.8 in2

Simplify.

1 100

10,000

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

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

a. The surface area of a triangular prism is 34 square inches. What is the surface area of a similar prism that is twice as large? b. RAISINS The world’s largest box of raisins has a surface area of 352 square feet. If a similar box is smaller than the largest box 1 by a scale factor of _ , what is its surface area? 48

Refer to the cubes on the previous page. Volume of Small Cube V=4·4·4

Volume of Large Cube 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.

Lesson 2C Similar Solids

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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? 1 V = 432 × _

(3)

3

Multiply by the cube of the scale factor.

1 V = 432 × _

Cube _.

V = 16 yd3

Simplify.

27

1 3

The volume of the new prism is 16 cubic yards.

c. 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?

HOCKEY The standard hockey puck measures as shown at the right. Find the surface area and volume of the giant puck at the left. Use 3.14 for π.

1.5 in.

1 in.

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

Real-World Link R TThe hockey puck that appears to be crashing into the side of the wall at Nationwide Arena in Columbus, Ohio, is about 40 times the actual size of a standard puck.

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.

d. ERASERS The dimensions of a rectangular eraser are 2.4 inches by 4.6 inches by 1 inch. Find the surface area and volume of a similar eraser that is 5 times as large.

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Measurement and Proportional Reasoning

Example 1

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?

Example 2

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 1 that is smaller by a scale factor of _ ? 100

Example 3

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 1 factor of _ ? 3

Example 4

4. ART STUDIO A sink with a sliding lid in Josh’s art studio measures 16 inches by 15 inches by 6 inches. A second sink used just for paintbrushes has a 1 similar shape and is smaller by a scale factor of _ . Find the surface area 2 and volume of the second sink.

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

Example 1

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. 6. The surface area of a triangular prism is 10.4 square meters. What is the 1 surface area of a similar solid that is smaller by a scale factor of _ ? 4

Example 2

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?

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? Example 3

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

Example 4

11. ARCHITECTURE The model of a new 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?

10 in.

28 in.

18 in.

Lesson 2C Similar Solids

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12. PUZZLES The world’s largest cube puzzle is in Knoxville, Tennessee. It measures 6 feet on each side. The scale factor between a standard cube 1 puzzle and largest puzzle is _ . Find the surface area and volume of the 24 standard cube puzzle. 13 Two spheres are similar in shape. The scale factor between the smaller _3 B sphere and the larger sphere is 4 . If the volume of the smaller sphere is R Real-World Link Cube Puzzles C 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.

126.9 cubic meters, what is the volume of the larger sphere? Determine whether each statement is always, sometimes, or never true. 14. Two prisms with equal bases are similar. 15. Similar solids have equal volumes. 16. Two cubes are similar. 17. A prism and pyramid are similar. 18. Find the missing measure for the pair of similar solids.

y ft

1.5 ft

7 ft 10.5 ft

3 ft

19. Two similar cylinders are shown.

x ft

18 cm

a. What is the ratio of their radii? 6 cm

b. What is the ratio of their surface areas? volumes? c. Find the surface area of Cylinder B. d. Find the volume of Cylinder A.

S.A. = 5,425.92 cm2

V = 1,130.4 cm3

Cylinder A

Cylinder B

20. MEASUREMENT Find the missing values in the table. Use 3.14 for π. Volume (cm3)

Surface Area (cm2)

Radius (cm)

Cylinder A

3.14

12.57

1

Cylinder B

84.82

Cylinder C

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Measurement and Proportional Reasoning

452.52

C 21. CHALLENGE A frustum is the solid left 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.

3 in. 6 in. 1.5 in.

frustum

3 in.

b. What is the volume of the smaller cone? the larger cone? Use 3.14 for π.

c. What is the ratio of the volume of the smaller cone to the volume of the larger cone? d. What is the volume of the frustum? 22. E WRITE MATH Explain what happens to the volume of a prism when its dimensions are tripled.

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

24. Two similar prisms have volumes of 4 cubic meters and 864 cubic meters, respectively. How many times larger is the second prism? F. 6 times G. 16 times

25 cm

5 A. _

3 25 B. _ 15

15 cm

25 C. _

H. 96 times I.

216 times

9 10 D. _ 6

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 2A) 26. SPORTS The table shows the dimensions of the fields used in various sports. (Lesson 1F)

Sport

Length (yards)

Width (yards)

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

Field hockey

60

100

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

Football

53_

120

Lacrosse

60

110

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

Soccer

70

115

1 3

Lesson 2C Similar Solids

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