When the dimensions of a solid increase by a factor of k, how does the surface area change? How does the volume change?
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1 8.4 Surface Areas and Volumes of Similar Solids Wen te dimensions of a solid increase by a factor of k, ow does te surface area cange? How does te volume cange? 1 ACTIVITY: Comparing Surface Areas and Volumes Work wit a partner. Copy and complete te table. Describe te pattern. Are te dimensions proportional? Explain your reasoning. a. Radius Heigt Surface Area Volume b. Geometry In tis lesson, you will identify similar solids. use properties of similar solids to find missing measures. understand te relationsip between surface areas of similar solids. understand te relationsip between volumes of similar solids. solve real-life problems. Radius Heigt Surface Area Volume 354 Capter 8 Volume and Similar Solids
2 2 ACTIVITY: Comparing Surface Areas and Volumes Work wit a partner. Copy and complete te table. Describe te pattern. Are te dimensions proportional? Explain. Mat Practice Repeat Calculations Wic calculations are repeated? How does tis elp you describe te pattern? Base Side Heigt Slant Heigt Surface Area Volume 3. IN YOUR OWN WORDS Wen te dimensions of a solid increase by a factor of k, ow does te surface area cange? 4. IN YOUR OWN WORDS Wen te dimensions of a solid increase by a factor of k, ow does te volume cange? 5. REPEATED REASONING All te dimensions of a prism increase by a factor of 5. a. How many times greater is te surface area? Explain b. How many times greater is te volume? Explain Use wat you learned about surface areas and volumes of similar solids to complete Exercise 3 on page 359. Section 8.4 Surface Areas and Volumes of Similar Solids 355
3 8.4 Lesson Lesson Tutorials Key Vocabulary similar solids, p. 356 Similar solids are solids tat ave te same sape and proportional corresponding dimensions. EXAMPLE 1 Identifying Similar Solids Cylinder B Wic cylinder is similar to Cylinder A? Cylinder A 5 m 3 m Ceck to see if corresponding dimensions are proportional. 6 m 4 m Cylinder A and Cylinder B Cylinder C Heigt of A Heigt of B = 4 3 Radius of A Radius of B = 6 5 Not proportional 7.5 m Cylinder A and Cylinder C 5 m Heigt of A Heigt of C = 4 5 Radius of A Radius of C = = 4 5 Proportional So, Cylinder C is similar to Cylinder A. EXAMPLE 2 Finding Missing Measures in Similar Solids Cone X 13 yd Cone Y Te cones are similar. Find te missing slant eigt. Radius of X Radius of Y = Slant eigt of X Slant eigt of Y 5 7 = 13 Substitute. 5 yd 7 yd 5 = 91 Cross Products Property = 18.2 Divide eac side by 5. Te slant eigt is 18.2 yards. Exercises Cylinder D as a radius of 7.5 meters and a eigt of 4.5 meters. Wic cylinder in Example 1 is similar to Cylinder D? 2. Te prisms at te rigt are similar. Find te missing widt and lengt. 11 in. 20 in. 8 in. 8 in. w 356 Capter 8 Volume and Similar Solids
4 Linear Measures r r w w r Surface Areas of Similar Solids Wen two solids are similar, te ratio of teir surface areas is equal to te square of te ratio of teir corresponding linear measures. Solid A a Solid B b Surface Area of A Surface Area of B = ( a b) 2 EXAMPLE 3 Finding Surface Area 6 ft Pyramid A Pyramid B Te pyramids are similar. Wat is te surface area of Pyramid A? Surface Area of A Surface Area of B ( = Heigt of A Heigt of B) 2 S 600 ( = 10) 6 2 Substitute. S 600 = S = Evaluate. Multiplication Property of Equality S = 216 Simplify. Surface Area 600 ft 2 Te surface area of Pyramid A is 216 square feet. Te solids are similar. Find te surface area of te red solid. Round your answer to te nearest tent cm 4 cm 8 m Surface Area 608 m 2 5 m Surface Area 110 cm 2 Section 8.4 Surface Areas and Volumes of Similar Solids 357
5 Volumes of Similar Solids Wen two solids are similar, te ratio of teir volumes is equal to te cube of te ratio of teir corresponding linear measures. Solid A a Solid B b Volume of A Volume of B = ( a b) 3 Study Tip EXAMPLE Original inal Tank Volume 2000 ft 3 Wen te dimensions of a solid are multiplied by k, te surface area is multiplied by k 2 and te volume is multiplied by k 3. 4 Finding Volume Te dimensions of te touc tank at an aquarium are doubled. Wat is te volume of te new touc tank? A 150 ft 3 B 4000 ft 3 C 8000 ft 3 D 16,000 ft 3 Te dimensions are doubled, so te ratio of te dimensions of te original tank to te dimensions of te new tank is 1 : 2. Original volume New volume 2000 V = ( V = 1 8 Original dimension = ( New dimension ) 3 2) 3 Substitute. Evaluate. 16,000 = V Cross Products Property Te volume of te new tank is 16,000 cubic feet. So, te correct answer is D. Exercises Te solids are similar. Find te volume of te red solid. Round your answer to te nearest tent cm 12 cm 3 in. Volume 9 in. 3 4 in. Volume 288 cm Capter 8 Volume and Similar Solids
6 8.4 Exercises Help wit Homework 1. VOCABULARY Wat are similar solids? 2. OPEN-ENDED Draw two similar solids and label teir corresponding linear measures. 9+(-6)=3 3+(-3)= 4+(-9)= 9+(-1)= 3. NUMBER SENSE All te dimensions of a cube increase by a factor of 3 2. a. How many times greater is te surface area? Explain. b. How many times greater is te volume? Explain. 1 Determine weter te solids are similar in. 2 in. 1 in. 9 in. 4 in. 4 in. 2 in. 2 in. 4 in. 1 in. 6 in. 3 in ft 6.5 ft 13 ft 12 ft 15 m 21 m 20 m 5 ft 5 ft 9 m 12 m 29 m Te solids are similar. Find te missing dimension(s) ft 9. 5 m c 12 m 13 m 6 m d 10 in. 7.5 m b Section 8.4 Surface Areas and Volumes of Similar Solids 359
7 Te solids are similar. Find te surface area S or volume V of te red solid. Round your answer to te nearest tent in. 15 in. 4 m Surface Area 336 m 2 6 m Surface Area 1800 in ft 21 mm 21 mm Volume 5292 mm 3 7 mm 7 mm Volume 7850 ft ERROR ANALYSIS Te ratio of te corresponding linear measures of two similar solids is 3 : 5. Te volume of te smaller solid is 108 cubic inces. Describe and correct te error in finding te volume of te larger solid. 15. MIXED FRUIT Te ratio of te corresponding linear measures of two similar cans of fruit is 4 to 7. Te smaller can as a surface area of 220 square centimeters. Find te surface area of te larger can. 108 V = ( 3 5 ) V = = V Te volume of te larger solid is 300 cubic inces. 16. CLASSIC MUSTANG Te volume of a 1968 Ford Mustang GT engine is 390 cubic inces. Wic scale model of te Mustang as te greater engine volume, a 1 : 18 scale model or a 1 : 24 scale model? How muc greater is it? 360 Capter 8 Volume and Similar Solids
8 17. MARBLE STATUE You ave a small marble statue of Wolfgang Mozart. It is 10 inces tall and weigs 16 pounds. Te original statue is 7 feet tall. a. Estimate te weigt of te original statue. Explain your reasoning. b. If te original statue were 20 feet tall, ow muc would it weig? 18. REPEATED REASONING Te largest doll is 7 inces tall. Eac of te oter dolls is 1 inc sorter tan te next larger doll. Make a table tat compares te surface areas and te volumes of te seven dolls. Wolfgang Mozart 19. Precision You and a friend make paper cones to collect beac glass. You cut out te largest possible tree-fourts circle from eac piece of paper. a. Are te cones similar? Explain your reasoning. b. Your friend says tat Friend s paper because your seet of paper is twice as large, your cone will old 8.5 in. exactly twice te volume of beac glass. 11 in. Is tis true? Explain your reasoning. Your paper 17 in. 11 in. Draw te figure and its reflection in te x-axis. Identify te coordinates of te image. (Section 2.3) 20. A(1, 1), B(3, 4), C(4, 2) 21. J( 3, 0), K( 4, 3), L( 1, 4) 22. MULTIPLE CHOICE Wic system of linear equations as no solution? (Section 5.4) A y = 4x + 1 y = 4x + 1 B y = 2x 7 y = 2x + 7 C 3x + y = 1 6x + 2y = 2 D 5x + y = 3 x + 5y = 15 Section 8.4 Surface Areas and Volumes of Similar Solids 361
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