Lesson 27 Solve Problems With Cylinders, Cones And Spheres

Develop Skills and Strategies Lesson 27 Part 1: Introduction CCLS Solve Problems with Cylinders, Cones, and Spheres 8.G.C.9 In Lesson 26, you learned about the formulas for the volumes of cylinders, cones, and spheres. In this lesson, you will use these formulas to solve problems. Janet is making homemade candles. She has two candle molds, a prism and a cylinder. Which mold uses the least amount of wax? 3 inches 8 inches 5 inches 8 inches 5 inches Explore It Explain how to use the area of the base to find the volume of the prism. What is the volume of wax needed for the candle in the shape of a prism? Explain how to find the volume of a cylinder. What is the shape of the base of the cylinder? What is the area of the base of the cylinder in terms of p? Write an expression for the volume of the cylinder in terms of p. What is the volume of the cylinder? (Use 3.14 for p and round your answer to the nearest tenth.) Which candle uses the least amount of wax? Explain. 226 L27: Solve Problems with Cylinders, Cones, and Spheres Curriculum Associates, LLC Copying is not permitted.

Part 1: Introduction Lesson 27 Find Out More The formula for the volume of a cylinder is related to the formula for the volume of a prism. In both cases, you multiply the area of the base times the height. The formula for the volume of a cylinder can be used to remember the formula for the volume of a cone with the same radius and height and the volume of a sphere with the same radius. The volume of a cone is 1 the volume of the cylinder and the volume of the sphere is 4 the volume of the cylinder. 3 ·· 3 ·· Volume of a cylinder 5 Area of the base ? height r 2 5 (pr ) h h 5 pr2h Volume of a cone 5 1 area of the base ? height r 3 ·· 5 1 pr2h 3 ·· h Volume of a sphere 5 4 pr2 ? r 3 ·· 5 4 pr3 3 ·· The “height” of a half-sphere is another radius, so h 5 r. r Reflect 1 How can you use the formula for the volume of a prism to remember the formula for the volume of a cylinder? How is the formula for the volume of a cone related to the formula for the volume of a cylinder? L27: Solve Problems with Cylinders, Cones, and Spheres Curriculum Associates, LLC Copying is not permitted. 227

Part 2: Modeled Instruction Lesson 27 Read the problem below. Then explore how to find the volume of each figure. Mark is buying a container to plant flowers in. His choices are a container in the shape of a cone with a height of 24 inches and a radius of 6 inches or one in the shape of a cylinder with a height of 8 inches and a diameter of 12 inches. Which container holds the least amount of soil? Picture It Make a sketch for each container. 6 inches 24 inches 12 inches 8 inches Model It Use the formula for the volume of a cone. Volume of a cone 5 1 area of the base height 3 ·· 5 1 pr2h 3 ·· 5 1 p (6)2 24 3 ·· Model It Use the formula for the volume of a cylinder. Volume of a cylinder 5 area of the base height 5 pr2h 5 p (6)2 8 228 L27: Solve Problems with Cylinders, Cones, and Spheres Curriculum Associates, LLC Copying is not permitted.

Part 2: Guided Instruction Lesson 27 Connect It Now you will compare the volumes of the containers on the previous page. 2 How does the radius of the cone compare to the radius of the cylinder? 3 How does the height of the cone compare to the height of the cylinder? 4 How does the formula for the volume of a cone compare to the formula for the volume of a cylinder? 5 Based on your answers to problems 2, 3, and 4, what do you think will be the ratio of the volume of the cone to the volume of the cylinder? Explain your reasoning. 6 Find the volume of each container. Write the volume in terms of p. cone 5 cylinder 5 7 Which container holds the least amount of soil? Try It Use what you just learned about volume to solve these problems. 8 What is the volume of wax in the candle shaped like a cylinder? Use 3.14 for p and round your answer to the nearest tenth. 3 inches 6 inches 4 inches 9 What is the volume of wax in the candle shaped like a cone? Use 3.14 for p and round your answer to the nearest tenth. L27: Solve Problems with Cylinders, Cones, and Spheres Curriculum Associates, LLC Copying is not permitted. 2 inches 229

Part 3: Modeled Instruction Lesson 27 Read the problem below. Then explore how to find the volume of each figure. A florist offers vases in three different sizes. Which vase holds the most water? Height: 20 cm Diameter: 10 cm Height: 30 cm Diameter: 10 cm Diameter: 10 cm Model It Use the formula for the volume of a cylinder. Volume of a cylinder 5 pr2h 5 p (5)2 20 Model It Use the formula for the volume of a cone. Volume of a cone 5 1 pr2h 3 ·· 5 1 p (5)2 30 3 ·· Model It Use the formula for the volume of a sphere. Volume of a sphere 5 4 pr3 3 ·· 5 4 p (5)3 3 ·· 230 L27: Solve Problems with Cylinders, Cones, and Spheres Curriculum Associates, LLC Copying is not permitted.

Part 3: Guided Instruction Lesson 27 Connect It Now you will compare the volumes of the containers on the previous page. Use 3.14 for p and round your answer to the nearest whole number. 10 How much water does the vase in the shape of a cylinder hold? 11 How much water does the vase in the shape of a cone hold? 12 How much water does the vase in the shape of a sphere hold? 13 Which vase holds the most water? 14 The florist buys new vases from a different supplier. The vases are the same shapes and have the same heights, but the diameter of each vase is 1 cm less than the original vases. Find the volume of water each of the new vases will hold. cone 5 cylinder 5 sphere 5 15 By what percent did the volume of each shape decrease? Round to the nearest whole percent. cylinder cone sphere 16 Look at the exponent of r in each of the three volume formulas. Explain why the sphere had the largest percent change in volume. Try It Use what you just learned about volume to solve this problem. Round to the nearest whole number. 17 Find the volume of the beach ball and the volume of the toy ball. How many times greater is the volume of the beach ball than the volume of the toy ball? L27: Solve Problems with Cylinders, Cones, and Spheres Curriculum Associates, LLC Copying is not permitted. 12 inches 6 inches 231

Part 4: Guided Practice Lesson 27 Study the model below. Then solve problems 18–20. Student Model The radius is half of the diameter, so the student used 3 for the radius of the sphere and 1.5 for the radius of the cylinder. Justin’s bird feeder is shown. About how much bird seed, rounded to the nearest whole number, does the bird feeder hold? Look at how you could show your work using the formulas for the volumes of cylinders and spheres. Volume of Sphere V 5 4 p (3)3 3 ·· 5 36 p 3 inches 8 inches 6 inches Volume of Cylinder V 5 p (1.5)2 8 5 18 p 36 p 1 18 p 5 54 p 54 ? 3.14 Pair/Share Why do you add the two volumes? What is the height of the punch if it is 1 inch less than the height of the punch bowl? 169.56 3 Solution: The feeder holds about 170 in. of bird seed. 18 A punch bowl is in the shape of a cylinder. The height of the punch bowl is 10 inches and the diameter of the punch bowl is 14 inches. What is the volume of punch you need to fill the punch bowl 1 inch from the top? (Use 3.14 for p and round your answer to the nearest whole.) Show your work. Pair/Share Why doesn’t it make sense to find the volume using the height of the entire punch bowl? 232 Solution: L27: Solve Problems with Cylinders, Cones, and Spheres Curriculum Associates, LLC Copying is not permitted.

Part 4: Guided Practice Lesson 27 19 The birthday candle shown has a radius of 5 mm. How much wax is needed to make the candle? Use 3.14 for p and round your answer to the nearest whole number. Show your work. I think that there are at least three different steps to solve this problem. 8 mm 20 mm Pair/Share Solution: 20 A container for a hanging plant is a half sphere with a diameter of 9 inches. About how much soil does the container hold? Circle the correct answer. A 191 in.3 B 382 in.3 C 1,526 in.3 D 3,052 in.3 How is this problem different from others you’ve seen in this lesson? What do you need to do to the formula for the volume of a sphere to find the formula for the volume of a half sphere? Rob chose C as the correct answer. How did he get that answer? Pair/Share What does Rob need to do with his answer to find the correct answer? L27: Solve Problems with Cylinders, Cones, and Spheres Curriculum Associates, LLC Copying is not permitted. 233

226 L27: Solve Problems with Cylinders, Cones, and Spheres Lesson 27 Part 1: Introduction Solve Problems with Cylinders, Cones, and Spheres In Lesson 26, you learned about the formulas for the volumes of cylinders, cones, and spheres. In this lesson, you will use these formulas to solve problems. Janet is making homemade candles.