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316 Chapter 7 Volumes of Solids
STATE
STANDARDS
MA.7.G.2.
S
7.4^ Volumes of Cones
Work with a partner. You can remember the volume formulas for all of the solids shown with just two concepts. Volumes of Prisms and Cylinders Volume = (Area of Base) ร (Height)
Volumes of Pyramids and Cones
Volume = (^1) โ 3 (Volume of Prism or Cylinder with same base and height)
Make a list of all the formulas you need to remember to find the area of a base. Talk about strategies for remembering these formulas.
11 ACTIVITY: Summarizing Volume Formulas
Prism Cone
Cylinder
Pyramid
Prism Prism
How can you remember the formulas for surface area and volume?
You discovered that the volume of a pyramid is one-third the volume of a prism that has the same base and same height. You can use a similar activity to discover that the volume of a cone is one-third the volume of a cylinder that has the same base and height.
Volume of a Cone = 1 โ 3 (Area of Base) ร (Height)
Height รข h
Area of Base รข B
Section 7.4 Volumes of Cones 317
Work with a partner. Make a list of the formulas for surface area that you studied in Chapter 6. Organize these formulas in a way similar to what you did in Activity 1.
Surface Area of a Right Prism =
Surface Area of a Right Pyramid =
Surface Area of a Right Cylinder =
Surface Area of a Right Cone =
33 ACTIVITY: Summarizing Surface Area Formulas
4. IN YOUR OWN WORDS How can you remember the formulas for surface area and volume? Write all of the surface area and volume formulas on a summary sheet. Make the list short so that you do not have to memorize many formulas.
Use what you learned about the volumes of cones to complete Exercises 4โ 6 on page 320.
Work with a partner. Think of a stack of paper. If you adjust the stack so that the sides are oblique (slanted), do you change the volume of the stack? If the volume of the stack does not change, then the formulas for volumes of right solids also apply to oblique solids.
22 ACTIVITY: Volumes of Oblique Solids
h = 4
B = 4 ฯ
h = 4
B = 4 ฯ
h = 5
B = 9 ฯ
h = 5
B = 9 ฯ Right cylinder Oblique cylinder Right cone Oblique cone
Section 7.4 Volumes of Cones 319
Find the volume V or height h of the cone. Round your answer to the nearest tenth. 1.
15 cm
V โ
6 cm (^) 2.
15 yd
h โ
Volume = 7200 yd 3
EXAMPLE 33 Real-Life Application
You must answer a trivia question before the sand in the timer falls to the bottom. The sand falls at a rate of 50 cubic millimeters per second. How much time do you have to answer the question? Use the formula for the volume of a cone to find the volume of the sand in the timer.
V = 1 โ 3 Bh Write formula.
1 โ 3 ฯ (10)^2 (24) Substitute.
= 800 ฯ โ 2512 Simplify. The volume of the sand is about 2512 cubic millimeters. To find the amount of time you have to answer the question, multiply the volume by the rate at which the sand falls.
2512 mm^3 ร 1 sec โ 50 mm 3 = 50.24 sec
You have about 50 seconds to answer the question.
3. WHAT IF? In Example 3, the sand falls at a rate of 60 cubic millimeters per second. How much time do you have to answer the question? 4. WHAT IF? In Example 3, the height of the sand in the timer is 12 millimeters and the radius is 5 millimeters. How much time do you have to answer the question?
30 mm
10 mm
24 mm
Exercises 4โ
7.4 Exercises
320 Chapter 7 Volumes of Solids
V =
1 โ 3
Bh
= 1 โ 3
( ๐ )(6)^2 (8)
= 96 ๐ m^3
8 m โ
6 m
10 cm
3 cm 4 cm
8 cm
Find the volume of the cone. Round your answer to the nearest tenth. 4.
2 in.
4 in.
6 m
3 m 6.
5 mm
10 mm
7. 2 ft^ 1 ft 8. 5 cm
8 cm
6 yd
9 yd
3 ft
7 ft 11.
5 in.
10 in.
12. 4 cm
8 cm
13. ERROR ANALYSIS Describe and correct the error in fi nding the volume of the cone. 14. GLASS The inside of each glass is shaped like a cone. Which glass can hold more liquid? How much more?
9 +(-6)=3 3 +(-3)= 4 +(-9)= 9 +(-1)=
1. VOCABULARY Describe the height of a cone. 2. WRITING Compare and contrast the formulas for the volume of a pyramid and the volume of a cone. 3. REASONING You know the volume of a cylinder. How can you find the volume of a cone with the same base and height?
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