Lesson 27 Introduction M.8.24 Solve roblems with Cylinders ...€¦ · Lesson 27 Solv roblem it...

Introduction

©Curriculum Associates, LLC Copying is not permitted.226

Use What You Know

Lesson 27Solve Problems with Cylinders, Cones, and Spheres

Lesson 27 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. She has two candle molds, a prism and a cylinder. Which mold uses the least amount of wax?

8 inches

5 inches5 inches

8 inches

3 inches

Use the math you already know to solve the problem.

a. Explain how to use the area of the base to find the volume of the prism.

b. What is the volume of wax needed for the candle in the shape of a prism?

c. Explain how to find the volume of a cylinder.

d. What is the shape of the base of the cylinder?

e. What is the area of the base of the cylinder in terms of p?

f. Write an expression for the volume of the cylinder in terms of p.

g. What is the volume of the cylinder? (Use 3.14 for p and round your answer to the

nearest tenth.)

h. Which candle uses the least amount of wax? Explain.

M.8.24

©Curriculum Associates, LLC Copying is not permitted. 227

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 ·· 3 the volume of the cylinder and the volume of the sphere

is 4 ·· 3 the volume of the cylinder.

Volume of a cylinder 5 Area of the base ? height

5 (pr2) • h

5 pr2h

Volume of a cone 5 1 ·· 3 • area of the base ? height

5 1 ·· 3 • pr2 ? h

Volume of a sphere 5 4 ·· 3 • pr2 ? r

5 4 ·· 3 pr3

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?

r

h

r

h

r

The “height” of a half sphere is another radius,so h = r.

Modeled and Guided Instruction

Learn About


Lesson 27


Solving Volume Problems

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

8 inches

12 inches

Model It Use the formula for the volume of a cone.

Volume of a cone 5 1 ·· 3 • area of the base • height

5 1 ·· 3 pr2h

5 1 ·· 3 p • (6)2 • 24

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

©Curriculum Associates, LLC Copying is not permitted. 229Lesson 27 Solve Problems with Cylinders, Cones, and Spheres

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 2–4, 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.

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.

4 inches

3 inches

6 inches

2 inches

Modeled and Guided Instruction

Learn About


Lesson 27


Comparing Volumes

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 cmDiameter: 10 cm

Height: 30 cmDiameter: 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 ·· 3 • pr2h

5 1 ·· 3 p • (5)2 • 30

Model It Use the formula for the volume of a sphere.

Volume of a sphere 5 4 ·· 3 pr3

5 4 ·· 3 p • (5)3


6 inches

12 inches

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.

cylinder 5 cone 5

sphere 5

15 By what percent did the volume of each shape decrease? Round to the nearest whole percent.

cylinder 5 cone 5 sphere 5

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. Show your work on a separate sheet of paper.

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?

Practice


Guided PracticeLesson 27



Study the example below. Then solve problems 18–20.

Example

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.

Solution

3 inches

6 inches

8 inches

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 number.)

Show your work.

Solution

Pair/ShareWhy do you add the two volumes?

Pair/ShareWhy doesn’t it make sense to find the volume using the height of the entire punch bowl?

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.

What is the height of the punch if it is 1 inch less than the height of the punch bowl?

Volume of Sphere Volume of Cylinder

V 5 4 ·· 3 p • (3)3 V 5 p • (1.5)2 • 8

5 36 p 5 18 p

36 p 1 18 p 5 54 p

≈ 54 ? 3.14

≈ 169.56

The feeder holds about 170 in.3 of bird seed.


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.

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

Rob chose C as the correct answer. How did he get that answer?

Pair/ShareHow is this problem different from others you’ve seen in this lesson?

Pair/ShareWhat does Rob need to do with his answer to find the correct answer?

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?

I think that there are at least three different steps to solve this problem.

20 mm

8 mm

Independent Practice

Practice


Lesson 27



Fido

18 cm

Solve the problems.

1 An automatic dog feeder in the shape of a cone has a height of 18 centimeters. If it holds about 8,308 cubic centimeters of dog food, approximately what is its radius?

A 7 cm

B 14 cm

C 21 cm

D 49 cm

2 A serving bowl has two parts, a half sphere that holds food and a cylindrical base that holds cold water to keep the food cold. Which expression represents the volume of water in the cylindrical base?

A pr2h

B pr2h 1 2 ·· 3 pr3

C pr2h 2 2 ·· 3 pr3

D pr2h 2 4 ·· 3 pr3

3 A sphere and a cone have the same volume, and each has a radius of 6 centimeters. What is the height of the cone?

The height of the cone is cm

r

h

Self Check


Go back and see what you can check off on the Self Check on page 159.

4 At a construction site, a crane is lifting water in cone-shaped container and transferring the water to a cylindrical storage tank.

10 meters8 meters

Water Container Storage Tank

4 meters

3 meters

What is the maximum number of full containers of water that can be transferred into the storage tank without the water overflowing the tank? Shade in the maximum number of containers.

Water Containers

5 Kristen gets a full scoop of frozen yogurt in a cone. The scoop is a perfect sphere with the same radius as the cone. She wonders if the entire volume of the frozen yogurt could fit completely inside the cone. What is the relationship between the volume of the cone and the volume of frozen yogurt? Is the volume of the cone great enough to fit all of the frozen yogurt? Explain. (Use 3.14 for p and round your answer to the nearest whole number.)

5 cm

11 cm

Lesson 27 Introduction M.8.24 Solve roblems with Cylinders ...€¦ · Lesson 27 Solv roblem it...

Documents

Transcript of Lesson 27 Introduction M.8.24 Solve roblems with Cylinders ...€¦ · Lesson 27 Solv roblem it...