CHAPTER

9Geometry and Measurement

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9.1 Systems of Linear Measurement9.2 Converting Units of Area9.3 More with Perimeter and Area9.4 Volume and Capacity9.5 Angles and Triangles9.6 Square Roots and the Pythagorean Theorem9.7 Weight, Mass, and Temperature9.8 Medical Applications

OBJECTIVES

9.4 Volume and Capacity

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a Find the volume of a rectangular solid, a cylinder, and a sphere.

b Convert from one unit of capacity to another.c Solve applied problems involving volume and

capacity.

9.4 Volume and Capacity

a Find the volume of a rectangular solid, a cylinder, and a sphere.

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The volume of an object is the number of unit cubes needed to fill it.

9.4 Volume and Capacity

VOLUME OF A RECTANGULAR SOLID

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The volume of a rectangular solid is found by multiplying length by width by height:

EXAMPLE

9.4 Volume and Capacity

a Find the volume of a rectangular solid, a cylinder, and a sphere.

2 Volume of a Safe.

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For office security, William purchases a safe with dimensions Find the volume of this rectangular solid.

9.4 Volume and Capacity

a Find the volume of a rectangular solid, a cylinder, and a sphere.

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Volumes are described in units such as cubic centimeters (cm3) and cubic inches (in3). Dimensional analysis is an excellent way of determining the correct units for an answer.• If measurements of length are added, use a one-dimensional unit of length.• If two measurements of length are multiplied, use a two-dimensional unit of area.• If three measurements of length are multiplied, use a three-dimensional unit of volume.

9.4 Volume and Capacity

a Find the volume of a rectangular solid, a cylinder, and a sphere.

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A rectangular solid is shown. Note that we can think of the volume as the product of the area of the base times the height: where B represents the area of the base.

9.4 Volume and Capacity

a Find the volume of a rectangular solid, a cylinder, and a sphere.

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Like rectangular solids, circular cylinders have bases of equal area that lie in parallel planes. The bases of circular cylinders are circular regions.

9.4 Volume and Capacity

VOLUME OF A CIRCULAR CYLINDER

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The volume of a circular cylinder is the product of the area of the base B and the height h:

EXAMPLE

Find the volume of this circular cylinder. Use 3.14 for

9.4 Volume and Capacity

a Find the volume of a rectangular solid, a cylinder, and a sphere.

3

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9.4 Volume and Capacity

a Find the volume of a rectangular solid, a cylinder, and a sphere.

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A sphere is the three-dimensional counterpart of a circle. It is the set of all points in space that are a given distance (the radius) from a given point (the center). The volume of a sphere depends on its radius.

9.4 Volume and Capacity

VOLUME OF A SPHERE

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The volume of a sphere of radius r is given by

EXAMPLE

The radius of a standard-sized bowling ball is 4.2915 in. Find the volume of a standard-sized bowling ball (disregarding the finger holes). Round to the nearest hundredth of a cubic inch. Use 3.14 for

9.4 Volume and Capacity

a Find the volume of a rectangular solid, a cylinder, and a sphere.

4 Bowling Ball.

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9.4 Volume and Capacity

b Convert from one unit of capacity to another.

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To answer a question like “How much soda is in the bottle?” we need measures of capacity.

9.4 Volume and Capacity

AMERICAN UNITS OF CAPACITY

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9.4 Volume and Capacity

b Convert from one unit of capacity to another.

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Fluid ounces, abbreviated fl oz, are often referred to as ounces, or oz.

EXAMPLE

9.4 Volume and Capacity

b Convert from one unit of capacity to another.

5

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Since we are converting from a smaller unit to a larger unit, we multiply by 1 using 1 gal in the numerator and 4 qt in the denominator:

EXAMPLE

9.4 Volume and Capacity

b Convert from one unit of capacity to another.

6

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The box with capacity units does not list how many ounces are in 1 gal. We convert gallons to quarts, quarts to pints, and pints to ounces, using the relationships given in the box.

EXAMPLE

9.4 Volume and Capacity

b Convert from one unit of capacity to another.

6

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Since we are converting from a larger unit to a smaller unit, we use substitution:

Thus, 9 gal = 1152 oz.

9.4 Volume and Capacity

b Convert from one unit of capacity to another.

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One unit of capacity in the metric system is a liter.

9.4 Volume and Capacity

METRIC UNITS OF CAPACITY

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9.4 Volume and Capacity

b Convert from one unit of capacity to another.

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The metric prefixes are also used with liters. The most common is milli-.

Title

9.4 Volume and Capacity

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9.4 Volume and Capacity

b Convert from one unit of capacity to another.

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A common unit for drug dosage is the milliliter (mL) or the cubic centimeter (cm3). The notation “cc” is also used for cubic centimeter, especially in medicine. The milliliter and the cubic centimeter represent the same measure of capacity.

Volumes for which quarts and gallons are used are expressed in liters. Large volumes are expressed using measures of cubic meters (m3).

Title

9.4 Volume and Capacity

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EXAMPLE

9.4 Volume and Capacity

b Convert from one unit of capacity to another.

8

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EXAMPLE

9.4 Volume and Capacity

c Solve applied problems involving volume and capacity.

9

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At a self-service gas station, 89-octane gasoline sells for 102.6¢ a liter. Estimate the price of 1 gal in dollars.

Since 1 liter is about 1 quart and there are 4 quarts in a gallon, the price of a gallon is about 4 times the price of a liter.

Thus, 89-octane gasoline sells for about $4.10 a gallon.