Rational Expressions and Functions Chapter 8 Rational Expressions and Functions

8.6 Variation

Write an equation expressing direct variation. Objectives Write an equation expressing direct variation. Find the constant of variation, and solve direct variation problems. Solve inverse variation problems. Solve joint variation problems. Solve combined variation problems. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Write an equation expressing direct variation. If the value of one quantity depends on the value of another quantity, we can often describe that relationship using variation. That is, in many applied situations, the ratio of one quantity to another quantity remains the same. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Write an equation expressing direct variation. In the first example, as the number of feet increases, the number of inches increases. Also, as the number of feet decreases, the number of inches decreases. Both of these are examples of direct variation. In the second example, As the number of hours increases, the number of miles increases; and as the number of hours decreases, the number of miles decreases. Copyright © 2010 Pearson Education, Inc. All rights reserved.

The Definition of Direct Variation If y varies directly as x, it is also correct to say that y is proportional to x. In direct variation, for k > 0, as the value of x increases, the value of y increases. Similarly, as the value of x decreases, the value of y decreases. In any variation, there are at least two variables and a constant of variation. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Direct Variation Problem Given x or y, solve for the other variable. Substitute known values: x = 8 and y = 24. Start with Solve for k. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Direct Variation Problem The pressure exerted by a liquid at given point varies directly as the depth of the point beneath the surface of the liquid. If a certain liquid exerts a pressure of 50 pounds per square foot at a depth of 10 feet, find the pressure at a depth of 40 feet. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Direct Variation Problem The pressure exerted by a liquid t a given point varies directly a the depth of the point beneath the surface of the liquid. If a certain liquid exerts a pressure of 50 pounds per square foot at a depth of 10 feet, find the pressure at a depth of 40 feet. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Direct Variation Problem The pressure exerted by a liquid t a given point varies directly a the depth of the point beneath the surface of the liquid. If a certain liquid exerts a pressure of 50 pounds per square foot at a depth of 10 feet, find the pressure at a depth of 40 feet. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Direct Variation as a Power One variable may be proportional to the power of another variable. Examples of direct variation as a power: Area of a circle varies directly as the square of its radius. The distance an object falls varies directly as the square of the time its falls. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Direct Variation as a Power Problem The kinetic energy of an object varies directly as the square of its velocity. If an object with a velocity of 24 meters per second has a kinetic energy of 19,200 joules, what is the velocity of an object with a kinetic energy of 76,800 joules? Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Direct Variation as a Power Problem The kinetic energy of an object varies directly as the square of its velocity. If an object with a velocity of 24 meters per second has a kinetic energy of 19,200 joules, what is the velocity of an object with a kinetic energy of 76,800 joules? Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Direct Variation as a Power Problem Copyright © 2010 Pearson Education, Inc. All rights reserved.

8.6 Variation Inverse Variation Two variables are said to vary inversely if one is a constant multiple of the reciprocal of the other. Example: For a fixed area, the length of a rectangle is inversely proportional to its width. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving an Inverse Variation Problem Suppose that the time it takes to paint a house depends inversely on the number of members in the paint crew. As the number of painters increases, the time it takes to paint the house decreases. If it takes a crew of 4 painters 24 hours to paint a house, how long will it take to paint the same house if 6 painters work on the job? Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving an Inverse Variation Problem If it takes a crew of 4 painters 24 hours to paint a house, how long will it take to paint the same house if 6 painters work on the job? Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving an Inverse Variation Problem If it takes a crew of 4 painters 24 hours to paint a house, how long will it take to paint the same house if 6 painters work on the job? Notice that as the number of painters increased from 4 to 6, the time to paint the house decreased from 24 to 16 hours. Copyright © 2010 Pearson Education, Inc. All rights reserved.

It is possible for one variable to depend on several others 8.6 Variation Joint Variation It is possible for one variable to depend on several others Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Combined Variation Problem Ohm’s Law says that the current, I, in a wire varies directly as the electromotive force, E, and inversely as the resistance, R. If I is 11 amperes when E is 110 volts and R is 10 ohms, find I if E is 220 volts and R is 11 ohms. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Combined Variation Problem Ohm’s Law says that the current, I, in a wire varies directly as the electromotive force, E, and inversely as the resistance, R. If I is 11 amperes when E is 110 volts and R is 10 ohms, find I if E is 220 volts and R is 11 ohms. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Solving a Combined Variation Problem Ohm’s Law says that the current, I, in a wire varies directly as the electromotive force, E, and inversely as the resistance, R. If I is 11 amperes when E is 110 volts and R is 10 ohms, find I if E is 220 volts and R is 11 ohms. Copyright © 2010 Pearson Education, Inc. All rights reserved.