THE NATUREOF MEASUREMENT

Copyright © Cengage Learning. All rights reserved.

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Copyright © Cengage Learning. All rights reserved.

9.3 Surface Area, Volume, and Capacity

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Surface Area

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Surface Area

Suppose you wish to paint a box whose edges are each 3 ft, and you need to know how much paint to buy.

To determine this you need to find the sum of the areas of all the faces—this is called the surface area.

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Example 1 – Find the surface area of a box

Find the amount of paint needed for a box with edges 3 ft.

Solution:

A box (cube) has 6 faces of equal area.

6 9 ft2 = 54 ft2

You need enough paint to cover 54 ft2.

Number of faces of cube

Area of each face

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Surface Area

In the next example, we consider a box without a top and a can with a bottom but not a top.

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Example 2 – Find the outside surface area

Find the outside surface area.

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Example 2 – Solution

a. We find the sum of the areas of all the faces:

Front: 30 50 = 1,500

Back: 1,500

Side: 80 30 = 2,400

Side: 2,400

Bottom: 80 50 = 4,000

Total: 11,800 cm2

Same as front

Sides are the same size.

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Example 2 – Solution

b. To find the surface area, find the area of a circle (the bottom) and think of the sides of the can as being “rolled out.”

The length of the resulting rectangle is the circumference of the can and the width of the rectangle is the height of the can.

cont’d

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Example 2 – Solution

Side: A = /w = (d)w = (6)(6) = 36

Bottom: A = r2 = (3)2 = 9

Surface area: 36 + 9 = 45 141.37167

The surface area is about 141 cm2.

cont’d

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Volume

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Volume

To measure area, we covered a region with square units and then found the area by using a mathematical formula.

A similar procedure is used to find the amount of space inside a solid object, which is called its volume. We can imagine filling the space with cubes.

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Volume

A cubic inch and a cubic centimeter are shown in Figure 9.12. If the solid is not a cube but is a box (called a rectangular parallelepiped) with edges ofdifferent lengths, the volume can be found similarly.

Figure 9.12

Common units of measuring volume

a. 1 cubic inch (1 cu in. or 1 in.3)

b. 1 cubic centimeter (1 cu cm, cc, or 1 cm3)

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Volume

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Example 4 – Find the volume of a box

Find the volume of a box that measures 4 ft by 6 ft by 4 ft.

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Example 4 – Solution

There are 24 cubic feet on the bottom layer of cubes.

Do you see how many layers of cubes will fill the solid? Since there are four layers with 24 cubes in each, the total is

4 24 = 96

The volume is 96 ft3.

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Volume

This example leads us to the following formula for the volume of a box.

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Example 5 – Find the volume of a solid

Find the volume of each solid.

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Example 5 – Solution

a. V = s3

= (10 cm)3

= (10 10 10) cm3

= 1,000 cm3

b. V = /wh

= (25 cm)(10 cm)(4 cm)

= (25 10 4) cm3

= 1,000 cm3

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Example 5 – Solution

c. V = /wh

= (11 in.)(7 in.)(3 in.)

= (11 7 3) in.3

Sometimes the dimensions for the volume we are finding are not all given in the same units.

cont’d

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Example 5 – Solution

In such cases, you must convert all units to a common unit. The common conversions are as follows:

1 ft = 12 in. To convert feet to inches, multiply by 12.

To convert inches to feet, divide by 12.

1 yd = 3 ft To convert yards to feet, multiply by 3.

To convert feet to yards, divide by 3.

1 yd = 36 in. To convert yards to inches, multiply by 36.

To convert inches to yards, divide by 36.

cont’d

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Capacity

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Capacity

One of the most common applications of volume involves measuring the amount of liquid a container holds, which we refer to as its capacity.

For example, if a container is 2 ft by 2 ft by 12 ft, it is fairly easy to calculate the volume:

2 2 12 = 48 ft3

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Capacity

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Capacity

Some capacity statements from purchased products are listed in Table 9.1.

Table 9.1

Capacities of Common Grocery Items, as Shown on Labels

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Example 7 – Measure the quantity of a liquid

Measure the amount of liquid in the measuring cup in Figure 9.14, both in the U.S. system and in the metricsystem.

Figure 9.14

Standard measuring cup with both metric and U.S. measurements

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Example 7 – Solution

Metric: 240 mL

U.S.: About 1 c

or 8 oz

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Capacity

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Example 8 – Capacity of a container

How much water would each of the following containers hold?

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Example 8 – Solution

a. V = 90 cm 80 cm 40 cm

= 288,000 cm3

Since each 1,000 cm3 is 1 liter,

This container would hold 288 liters.

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Example 8 – Solution

b. V = 7 in. 22 in. 6 in.

= 924 in.3

Since each 231 in.3 is 1 gallon,

This container would hold 4 gallons.

cont’d

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Example 8 – Solution

Note that there is no need to estimate part a since the arithmetic is fairly easy.

However, for part b, the estimate might be

cont’d