28 LESSON Problem Solving: Volume READ Soup Can To the right is a diagram of a soup can. To the nearest tenth of a centimeter, what is the volume of the can? 8 cm The can looks like a, so use that volume formula. From Math Tool: Volume Formulas: V 5 r 2 h, where <.14, r is the radius, and h is the height. 10 cm The diagram shows that the diameter is 8 cm. The height is The formula requires the radius, and the radius is half the diameter. r 5 d 2 5 8 2 5 cm Substitute and simplify. V 5 r 2 h.14 ( ) 2 cm. Estimate to check that your answer is reasonable. For example, use instead of.14 for. ( ) 2 Is that estimate close to your answer? That estimate is to my answer, so my answer reasonable. The approximate volume of the soup can is cubic centimeters. 150 Domain 4: Geometry
read Carnival Treats At the school carnival, popcorn is given out in paper treat cones, like the one shown to the right. Approximately how many cubic inches can each cone hold? plan The shape is given. It is a cone. in. 5 in. Volume is measured in cubic units, so use the formula for finding the volume of a. From Math Tool: Volume Formulas: V 5 1 r 2 h, where <.14, r is the radius, and h is the height. solve The diagram shows that the height is 5 inches and the diameter is inches. The formula requires the radius, and the radius is the diameter. r 5 Substitute and simplify. V 5 1 r 2 h 1.14 ( ) 2 5 check Check that you answered the question that was asked. The question asked you to determine approximately how many cubic inches each cone can hold. Is your answer in cubic inches? Did you use an approximation for? The cone can hold approximately cubic inches. Lesson 28: Problem Solving: Volume 151
READ Beach Ball A beach ball has a diameter of 14 inches. If the beach ball is fully inflated, about how many cubic inches of air will it hold? 14 in. Volume is measured in cubic units, so use the formula for finding the volume of a. From Math Tool: Volume Formulas: V 5 4 r, where <.14 and r is the radius. The diagram shows that the is 14 inches. The formula requires the radius, and the radius is 1 2 the. r 5 Since the radius is, using 22 7 for will make the computation simpler. Substitute and simplify. V 5 4 r 4 22 7 ( ) Estimate to check that your answer is reasonable. Even though you didn t use.14 for, you can still use for to check your answer. 4 ( ) Is that estimate close to your answer? That estimate is to my answer, so my answer reasonable. The fully inflated beach ball will hold about cubic inches of air. 152 Domain 4: Geometry
Tennis Balls in a Can READ A cylindrical can holds three tennis balls snugly, one directly above another. If the radius of a tennis ball is.4 centimeters, what volume of the can is occupied by the air outside the tennis balls? The diagram shows spheres inside a cylinder. So, use the formula for finding the volume of a sphere, which is V 5 4 r. You will also use the formula for finding the volume of a cylinder, which is: V 5 The height will equal diameters, or times.4 cm. To find the air that is inside the cylinder, but outside the tennis balls, subtract the combined volume of the from the volume of the cylinder. Find the volume of one tennis ball, in terms of. V 5 4 (.4) cm Find the height of the cylinder: h 5 6r 5 6.4 5 cm Substitute that and other values into the formula for finding the volume of a cylinder. Find the volume in terms of. V 5 cm Subtract to find the volume you need. V 5 (volume of cylinder) 2 ( volume of one tennis ball) cm Use Math Tool: Volume Formulas. Are the formulas you used correct? The volume of the can that is occupied by the air outside the tennis balls is about cubic centimeters. Lesson 28: Problem Solving: Volume 15
Practice Use the 4-step problem-solving process to solve each problem. 1. READ Erika bought a plastic run-around ball for her guinea pig. diameter If the volume of the ball is 288 cubic inches, what is the diameter of the ball? 154 Domain 4: Geometry
2. About how many cubic meters of water will this swimming pool hold if it is filled to capacity? 6 m 2 m. A cone-shaped paper water cup has a radius of 4 centimeters and a volume of 48 cubic centimeters. What is the height of the cup? 4 cm h 4. A silo of a barn consists of a cylinder capped by a hemisphere (half-sphere). The height of the cylinder is 60 feet and its diameter is 0 feet. What is the approximate volume of the silo? 60 ft 0 ft Lesson 28: Problem Solving: Volume 155