Foundations of Math Section.5 Volume of Similar Figures 47.5 Volume of Similar Figures The cubes shown below are similar. The corresponding sides are in a ratio of :. What is the ratio of the volumes? The volume is 8 times larger by doubling the sides. Thus, we can state the following: Volume of large cube Volume of small cube = = 8 If the scale factor of two similar solids is a : b, then the ratio of the volumes is a : b. Example The ratio of similarity of two figures is 5. Find the ratio of their a) perimeters b) areas c) volumes a) perimeter of a : b is a : b, thus 5: b) area of a : b is a : b thus 5 : = 5: 4 c) volumes of a : b is a : b thus 5 : = 5:8
48 Chapter Rates and Scale Factor Foundations of Math Example The volume of a cone is V = r h. What happens to the volume of the cone if a) the height is doubled? b) the radius is doubled? c) both the height and radius are doubled? a) scale factor = ( ) r h r h = h h =, thus the volume is doubled. b) scale factor = c) scale factor = larger. r ( )h r h ( ) ( h) r r h ( = r ) = 4r = 4, thus the volume is quadrupled. r r ( = r ) h = 8r h r h r h = 8, thus the volume is 8 times Example Two similar brass statues are made of Carl Gauss, considered by many the world s greatest mathematician. The shoe on one statue is 9 cm long, and the other 5 cm long. If the volume of the smaller statue is 6 cm, what is the volume of the larger statue? 9 5 reduces to 5, thus V = l V l 6 = V 5 6 = 7 V 5 7V = 5 6 5 6 V = 7 V = 000 The larger statue would have a volume of000 cm.
Foundations of Math Section.5 Volume of Similar Figures 49 Example 4 The surface areas of two spheres are 6 cm and44 cm. What is the ratio of their volumes? Surface area has ratio S = l S l, volume has ratio V = l V l 6 44 = l l 4 = l l = l l V V = V V = 8 The ratio of their volumes is :8. Example 5 A can of tennis balls usually holds balls. What is the ratio of air in the can to volume of the can? 6r h Volume formula for cylinder is V = r h Volume formula for a sphere is V = 4 r balls would have a height of 6 radiis h = 6r Thus, volume of can is V = r ( 6r)= 6 r Volume of can minus volume of tennis balls = volume of air 6 r 4 r = r Scale factor = Volume of air Volume of can = r 6 r = 6 = The can is air.
50 Chapter Rates and Scale Factor Foundations of Math.5 Exercise Set. Complete the table for spheres: S = 4 r, V = 4 r Radius Area 6 64 Volume 4 6 48. Complete the table for two similar cones: V = r h Ratio of Height : Ratio of Base Area 4:9 Ratio of Volume 64:5. Complete the table for two similar pyramids: V = area of base height = ( A b )( h) (Plane DEF to base of pyramid ABCD) Scale Factor : Ratio of Area of Base 4:5 A Ratio of Volume of Smaller Pyramid to Volume of Larger Pyramid 7:64 B D F E C Ratio of Volume of Small Pyramid to Volume of Frustum (bottom part of pyramid) 8:9 D
Foundations of Math Section.5 Volume of Similar Figures 5 4. Two spheres of the same density have a surface area of 6 cm and 6 cm. If the small ball weighs 6 kg, what does the larger sphere weigh? 5. If a ball of yarn with a 6-in diameter cost $4 and an 8-inch diameter ball costs $8, which is the better buy? 6. A toy dump truck is a scale of :50 of an actual dump truck. If the toy dump truck will hold 5 cm of material, how many cubic metres will the actual dump truck hold? 7. The area of the base of two prisms is in a ratio of 4:9. If the heights are the same, what is the ratio of the volume?
5 Chapter Rates and Scale Factor Foundations of Math 8. Three tennis balls are packed tightly in a rectangular box. 9. If a sphere and a cone have the same radius and volume, what must be the height of the cone in terms of the radius? What is the ratio of the volume of the tennis balls to the volume of the box? 0. Two similar cones have lateral areas of 7 and 48. What is the ratio of their volumes?. A cylinder has its radius doubled and its height halved. What is the ratio of the volume of the original cylinder to the changed cylinder?
Foundations of Math Section.5 Volume of Similar Figures 5. The base area of two similar pyramids is 64 to 400. Find the ratio of their volumes.. A sphere of radius r is inscribed in a cone with diameter AC = AB = BC. B Find the ratio of the volume of the sphere to the volume of the cone. A C 4. Two identical cylinders are tightly packed side by side in a box cm by 6 cm and 4 cm height. Find the ratio of the volume of the box to the volume of the cylinders. 5. A solid ball of string has a diameter of yard. If one cubic inch of string weighs 0.05 pounds, how much does the ball of string weigh? ( yard = feet; foot = inches)