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672 Chapter 15 Volume and Similar Solids
15.1 Volumes of Cylinders
How can you fi nd the volume of a cylinder?
Work with a partner. a. Find the area of the face of a coin. b. Find the volume of a stack of a dozen coins. c. Write a formula for the volume of a cylinder.
11 ACTIVITY: Finding a Formula Experimentally
Work with a partner. You are planning to make and sell three different sizes of cylindrical candles. You buy 1 cubic foot of candle wax for $20 to make 8 candles of each size. a. Design the candles. What are the dimensions of each size of candle? b. You want to make a profit of $100. Decide on a price for each size of candle. c. Did you set the prices so that they are proportional to the volume of each size of candle? Why or why not?
22 ACTIVITY: Making a Business Plan
height â h
area of base â B
Geometry In this lesson, you will ● (^) find the volumes of cylinders. ● (^) find the heights of cylinders given the volumes. ● (^) solve real-life problems.
Section 15.1 Volumes of Cylinders 673
Work with a partner. Use the diagram to describe how you can find the volume of a small object.
33 ACTIVITY: Science Experiment
Work with a partner. a. Just by looking at the two cylinders, which one do you think has the greater volume? Explain your reasoning. b. Find the volume of each cylinder. Was your prediction in part (a) correct? Explain your reasoning.
44 ACTIVITY: Comparing Cylinders
5. IN YOUR OWN WORDS How can you find the volume of a cylinder? 6. Compare your formula for the volume of a cylinder with the formula for the volume of a prism. How are they the same?
Use what you learned about the volumes of cylinders to complete Exercises 3–5 on page 676.
5
10
15
20
25
3
4
9
2
“Here’s how I remember how to find the volume of any prism or cylinder.” “Base times tall, will fill
‘ em all.”
Consider Similar Problems How can you use the results of Activity 1 to fi nd the volumes of the cylinders?
Math Practice
Section 15.1 Volumes of Cylinders 675
EXAMPLE 33 Real-Life Application
How much salsa is missing from the jar? The empty space in the jar is a cylinder with a height of 10 − 4 = 6 centimeters and a radius of 5 centimeters.
V = Bh Write formula for volume. = π (5)^2 (6) Substitute. = 150 π ≈ 471 Use a calculator.
So, about 471 cubic centimeters of salsa are missing from the jar.
EXAMPLE 44 Real-Life Application
About how many gallons of water does the watercooler bottle contain? (1 ft^3 ≈ 7.5 gal)
A 5.3 gallons B 10 gallons C 17 gallons D 40 gallons
Find the volume of the cylinder. The diameter is 1 foot. So, the radius is 0.5 foot.
V = Bh Write formula for volume. = π (0.5)^2 (1.7) Substitute. = 0.425 π ≈ 1.3352 Use a calculator.
So, the bottle contains about 1.3352 cubic feet of water. To find the number of gallons it contains, multiply by the conversion factor 7.5 gal — 1 ft^3
1.3352 ft^3 × 7.5 gal — 1 ft^3 ≈ 10 gal
The watercooler bottle contains about 10 gallons of water. So, the correct answer is B.
3. WHAT IF? In Example 3, the height of the salsa in the jar is 5 centimeters. How much salsa is missing from the jar? 4. A cylindrical water tower has a diameter of 15 meters and a height of 5 meters. About how many gallons of water can the tower contain? (1 m^3 ≈ 264 gal)
Exercise 12
1 ft
1.7 ft
10 cm
4 cm
5 cm
15.1 Exercises
9 +(-6)=3 3 +(-3)= 4 +(-9)= 9 +(-1)=
676 Chapter 15 Volume and Similar Solids
1. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
How much does it take to fill the cylinder?
What is the capacity of the cylinder?
How much does it take to cover the cylinder?
How much does the cylinder contain?
2. REASONING Without calculating, which of the solids has the greater volume? Explain.
Help with Homework
5 cm
12 cm
8 in.
8 in.
8 in.
8 in.
8 in.
Find the volume of the cylinder. Round your answer to the nearest tenth.
3. (^) 9 ft
6 ft
3 m
3 m (^) 5. (^) 7 ft
5 ft
6. 5 ft
10 ft
3 mm
10 mm 8. (^) 2 ft 1 ft
9. (^) 7 in.
6 in.
5 m
15 m 11.
12. SWIMMING POOL A cylindrical swimming pool has a diameter of 16 feet and a height of 4 feet. About how many gallons of water can the pool contain? Round your answer to the nearest whole number. (1 ft 3 ≈ 7.5 gal)
16 cm
8 cm
