Multi-Part Lesson 9-2 Similar Solids PART Main Idea Solve problems involving similar solids. New Vocabulary similar solids glencoe.com A B C Changes in Dimensions MODELS Stephen is creating a model of the Washington Monument for history class. The model will be _ of the monument s 00 actual size.. 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 2 with the actual monument. or _ of the actual area (_ 00 ) 0,000 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) =C09-029A-8964 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. 528 Chapter 9 Measurement and Proportional Reasoning 526_5_C9L2_8964.indd 528 7/27/09 2:04:29 PM
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 times as large? S.A. = 78 2 Multiply by the square of the scale factor. S.A. = 78 9 Square. S.A. = 702 cm2 MODELS Refer to page 528. The surface area of the exposed portion of the pyramid atop the Washington Monument is 4,02 square feet. What is the surface area in square inches, to the nearest tenth, of the pyramid on Stephen s model? S.A. = 4,02 _ Multiply by the square of the scale factor. S.A. = 4,02 _ Square _. S.A. = 0.402 ft2 ( 00 )2 00 0,000 Real-World Link It takes about 4_ pounds of 2 fresh grapes to make one pound of raisins. 2 in. 2 in. S.A. = 0.402 ft ft _ _ Convert to inches. S.A. = 57.8 in2 ft ft 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 52 square feet. If a similar box is smaller than the Kingsburg box by a scale factor _ of _, what is its surface area? ft2 or 0.5 ft2 48 72 The volumes of similar solids are also related. Refer to the cubes on page 528. 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) = 2(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 9-2 Similar Solids 526_5_C9L2_8964.indd 529 529 7/0/09 :04:24 PM
Volume of Similar Solids A triangular prism has a volume of 42 cubic yards. If the prism is reduced to one third its original size, what is the volume of the new prism? () V = 42 _ Multiply by the cube of the scale factor. V = 42 _ Cube _. V = 6 yd 27 The volume of the new prism is 6 cubic yards. b. A square pyramid has a volume of 52 cubic centimeters. What is the volume of a square pyramid with dimensions one fourth of the original? 8 cm HOCKEY The standard hockey puck measures as shown at the right. Find the surface area and volume of the giant puck at the left..5 in. in. Find the volume and surface area of the standard puck first. V = πr2h S.A. = 2(πr2) + 2πrh (.4)(.5)2() 2(.4)(.5)2 + 2(.4)(.5)() 7.065 in 4. + 9.42 $ " 2.55 in2 Real-World Link The 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. c. 25,000 in2; 6,562,500 in 50 Find the volume and surface area of the giant puck using the scale factor. V = V(40) S.A. = S.A.(40)2 = (7.065)(40) = (2.55)(40)2 = 452,60 in = 7,680 in2 The giant hockey puck has a volume of 452,60 cubic inches and a surface area of 7,680 square inches. c. The dimensions of a rectangular solid are 0 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. Chapter 9 Measurement and Proportional Reasoning 526_5_C9L2_8964.indd 50 7/0/09 :04:27 PM
Example Example 2 Example (p. 50) Example 4 (p. 50). The surface area of a rectangular prism is 5 square inches. What is the surface area of a similar solid that has been enlarged by a scale factor of 7?,75 in 2 2. MODELS The surface area of a ship s hull is about,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 _?. m2 00. 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 _? 5.7 cm 4. ART STUDIO The sink in Josh s art studio measures 6 inches by 5 inches by 6 inches. A second sink used just for paint brushes has a similar shape and is smaller by a scale factor of _. Find the surface area and volume of 2 the second sink. 2 in 2 ; 80 in = Step-by-Step Solutions begin on page R. Extra Practice begins on page EP2. Example Example 2 Example (p. 50) 5. The surface area of a rectangular prism is,00 square inches. Find the surface area of a similar solid that is larger by a scale factor of.,700 in 2 6. The surface area of a triangular prism is 0.4 square meters. What is the surface area of a similar solid that is smaller by a scale factor of _? 0.65 m2 4 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.4? 548.8 in 2 8. DISPLAYS A glass display box has a surface area of 78 square inches. How many square inches of glass are used to create a glass display box with dimensions one-half the original? 94.5 in 2 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? 9 mm 0. A triangular prism has a volume of 50 cubic meters. If the dimensions are tripled, what is the volume of the new prism? 9,450 m Example 4 (p. 50). ARCHITECTURE The model of a new apartment building is shown. The architect plans for the building to be 44 times the size of the model. What will be the surface area and volume of the new building when it is completed? 25,72 ft 2 ; 8,709,20 ft 8 in. 0 in. 28 in. Lesson 9-2 Similar Solids 5 526_5_C9L2_8964.indd 5 7/0/09 :04:29 PM
2. 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. Two spheres are similar in shape. The scale factor between the smaller _ B sphere and the larger sphere is 4. If the volume of the smaller sphere is 26.9 cubic meters, what is the volume of the larger sphere? 00.8 m Real-World Link Cube Puzzles The world s largest automated cube puzzle solves itself every 0 seconds. The record time a standard cube puzzle was solved in 2008 was 7.08 seconds. Determine whether each statement is always, sometimes, or never true. 4. Two prisms with equal bases are similar. sometimes 5. Similar solids have equal volumes. sometimes 6. Two cubes are similar. always 7. A prism and pyramid are similar. never 8. Find the missing measure for the pair of similar solids. x = 2, y = 5.25 9b. surface area, 9 : ; volume, 27 : 9. Two similar cylinders are shown. a. What is the ratio of their radii? : y.5 ft 7 ft 0.5 ft ft Cylinder A Cylinder B 8 cm C09-05A-8964 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. 20. 4.7 mi/h; 5.68 mi/h; 6.8 mi/h; 8.42 mi/h x 6 cm S.A. = 5,425.92 cm2 0,520.8 cm V = 0.4 cm 20. GRAPHIC NOVEL Refer to the graphic novel frame below. Use the distances d and times you converted in Lesson 9-D andc09-06a-8964 the formula r = _ to t 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-D. 52 Chapter 9 Measurement and Proportional Reasoning 526_5_C9L2_8964.indd 52 7/0/09 :04:4 PM
C 2a. Yes, the ratios 6 and.5 are equal. 2. 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. b. What is the volume of the smaller cone? the larger cone? 7.065 in ; 56.52 in frustum 6 in. in. in..5 in. c. What is the ratio of the volume of the smaller cone to the volume of the larger cone? :8 d. What is the volume of the frustum? 49.455 in 22. Explain what happens to the volume of a prism when its dimensions are tripled. The volume is 27 times greater. Practice 2. For the similar pyramids, find the ratio of the surface area of the larger pyramid to the smaller pyramid. C A. B. 5_ 25_ 5 25 cm 5 cm C. 25_ 9 D. _ 0 6 24. Two similar prisms have volumes of 4 cubic meters and 864 cubic meters, respectively. How many times larger is the second prism? F F. 6 times larger G. 6 times larger H. 96 times larger I. 26 times larger 25. ART Julianna is making a clay figurine of a dog. The dog is 75 centimeters tall. If she uses a scale of centimeter = 0 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 in various sports. (Lesson 9-F) a. 54,000 ft 2 b.,050 ft 2 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 4,560 square feet, about how many acres are all four fields combined? about 5.6 acres Sport Length (yards) Width (yards) Field hockey 60 00 Football 5 _ 20 Lacrosse 60 0 Soccer 70 5 Lesson 9-2 Similar Solids 5 526_5_C9L2_8964.indd 5 7/0/09 :04:6 PM