Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Geometry and Measurement 9 9.1 Systems of Linear Measurement 9.2 Converting Units of Area 9.3 More with Perimeter and Area 9.4 Volume and Capacity 9.5 Angles and Triangles 9.6 Square Roots and the Pythagorean Theorem 9.7 Weight, Mass, and Temperature 9.8 Medical Applications Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 2

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity 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. c Determine whether an ordered pair is a solution of an equation with two variables. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 3

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity a Find the volume of a rectangular solid, a cylinder, and a sphere. The volume of an object is the number of unit cubes needed to fill it. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 4

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity VOLUME OF A RECTANGULAR SOLID The volume of a rectangular solid is found by multiplying length by width by height: a Find the volume of a rectangular solid, a cylinder, and a sphere. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 5

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity a Find the volume of a rectangular solid, a cylinder, and a sphere. 2 Volume of a Safe. For office security, William purchases a safe with dimensions Find the volume of this rectangular solid. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 6

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity a Find the volume of a rectangular solid, a cylinder, and a sphere. 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. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 7

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity a Find the volume of a rectangular solid, a cylinder, and a sphere. 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. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 8

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity a Find the volume of a rectangular solid, a cylinder, and a sphere. Like rectangular solids, circular cylinders have bases of equal area that lie in parallel planes. The bases of circular cylinders are circular regions. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 9

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity VOLUME OF A CIRCULAR CYLINDER The volume of a circular cylinder is the product of the area of the base B and the height h: a Find the volume of a rectangular solid, a cylinder, and a sphere. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 10

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity a Find the volume of a rectangular solid, a cylinder, and a sphere. 3 Find the volume of this circular cylinder. Use 3.14 for Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 11

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity a Find the volume of a rectangular solid, a cylinder, and a sphere. 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. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 12

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity VOLUME OF A SPHERE a The volume of a sphere of radius r is given by Find the volume of a rectangular solid, a cylinder, and a sphere. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 13

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity a Find the volume of a rectangular solid, a cylinder, and a sphere. 4 Bowling Ball. 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 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 14

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. To answer a question like “How much soda is in the bottle?” we need measures of capacity. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 15

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity AMERICAN UNITS OF CAPACITY b Convert from one unit of capacity to another. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 16

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. Fluid ounces, abbreviated fl oz, are often referred to as ounces, or oz. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 17

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. 5 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: Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 18

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. 6 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. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 19

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. 6 Since we are converting from a larger unit to a smaller unit, we use substitution: Thus, 9 gal = 1152 oz. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 20

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. One unit of capacity in the metric system is a liter. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 21

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity METRIC UNITS OF CAPACITY b Convert from one unit of capacity to another. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 22

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. The metric prefixes are also used with liters. The most common is milli-. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 23

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 24

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. 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). Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 25

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 26

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity b Convert from one unit of capacity to another. 8 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 27

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.4 Volume and Capacity c Solve applied problems involving volume and capacity. 9 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. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 28