Copyright © 2006 Pearson Education, Inc Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Rational Expressions, Equations, and Functions 6 Rational Expressions, Equations, and Functions 6.1 Rational Expressions and Functions: Multiplying and Dividing 6.2 Rational Expressions and Functions: Adding and Subtracting 6.3 Complex Rational Expressions 6.4 Rational Equations 6.5 Solving Applications Using Rational Equations 6.6 Division of Polynomials 6.7 Synthetic Division 6.8 Formulas, Applications, and Variation Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Formulas, Applications, and Variation 6.8 Formulas, Applications, and Variation Formulas Direct Variation Inverse Variation Joint and Combined Variation Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Formulas Formulas occur frequently as mathematical models. Many formulas contain rational expressions, and to solve such formulas for a specified letter, we proceed as when solving rational equations. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example In a hydraulic system, a fluid is confined to two connecting chambers. The pressure in each chamber is the same and is given by finding the force exerted (F) divided by the surface area (A). Therefore, we know Solve for A2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution Multiplying both sides by the LCD Simplifying by removing factors Dividing both sides by F1 This formula can be used to calculate A2 whenever A1, F2, and F1 are known. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Solve a Rational Equation for a Specified Variable If necessary, multiply both sides by the LCD to clear fractions. Multiply, as needed, to remove parentheses. Get all terms with the specified variable alone on one side. Factor out the specified variable if it is in no more than one term. Multiply or divide on both sides to isolate the specified variable. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Variation To extend our study of formulas and functions, we now examine three real-world situations: direct variation, inverse variation, and combined variation. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Direct Variation A mass transit driver earns $17 per hour. In 1 hr, $17 is earned. In 2 hr, $34 is earned. In 3 hr, $51 is earned, and so on. This gives rise to a set of ordered pairs: (1, 22), (2, 34), (3, 51), (4, 68), and so on. Note that the ratio of earnings E to time t is 17/1 in every case. If a situation gives rise to pairs of numbers in which the ratio is constant, we say that there is direct variation. Here earnings vary directly as the time: We have E/t = 17, so E = 17t, or using function notation, E(t) = 17t. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Direct Variation When a situation gives rise to a linear function of the form f (x) = kx, or y = kx, where k is a nonzero constant, we say that there is direct variation, that y varies directly as x, or that y is proportional to x. The number k is called the variation constant, or constant of proportionality. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find the variation constant and an equation of variation if y varies directly as x, and y = 15 when x = 3. Solution We know that (3, 15) is a solution of y = kx. Therefore, Substituting Solving for k The variation constant is 5. The equation of variation is y = 5x. The notation y(x) = 5x or f (x) = 5x is also used. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Inverse Variation To see what we mean by inverse variation, suppose it takes one person 8 hours to paint the baseball fields for the local park district. If two people do the job, it will take only 4 hours. If three people paint the fields, it will take only 2 and 2/3 hours, and so on. This gives rise to pairs of numbers, all have the same product: (1, 8), (2, 4), (3, 8/3), (4, 2), and so on. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Inverse Variation Note that the product of each pair of numbers is 8. Whenever a situation gives rise to pairs of numbers for which the product is constant, we say that there is inverse variation. Since pt = 8, the time t, in hours, required for the fields to be painted by p people is given by t = 8/p or, using function notation, t(p) = 8/p. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Inverse Variation When a situation gives rise to a rational function of the form f (x) = k/x, or y = k/x, where k is a nonzero constant, we say that there is inverse variation, that is y varies inversely as x, or that y is inversely proportional to x. The number k is called the variation constant, or constant of proportionality. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example The time, t, required to empty a tank varies inversely as the rate, r, of pumping. If a pump can empty a tank in 90 minutes at the rate of 1080 kL/min, how long will it take the pump to empty the same tank at the rate of 1500 kL/min? Solution 1. Familiarize. Because of the phrase “ . . . varies inversely as the rate, r, of pumping,” we express the amount of time needed to empty the tank as a function of the rate: t(r) = k/r Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution (continued) 2. Translate. We use the given information to solve for k. Then we use that result to write the equation of variation. Using function notation Replacing r with 1080 Replacing t(90) with 1080 Solving for k, the variation constant The equation of variation is t(r) = 97,200/r. This is the translation. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution (continued) 3. Carry out. To find out how long it would take to pump out the tank at a rate of 1500 mL/min, we calculate t(1500). t = 64.8 when r = 1500 4. Check. We could now recheck each step. Note that, as expected, as the rate goes up, the time it takes goes down. 5. State. If the pump is emptying the tank at a rate of 1500 mL/min, then it will take 64.8 minutes to empty the entire tank. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Joint and Combined Variation When a variable varies directly with more than one other variable, we say that there is joint variation. For example, in the formula for the volume of a right circular cylinder, V = πr2h, we say that V varies jointly as h and the square of r. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Joint Variation y varies jointly as x and z if, for some nonzero constant k, y = kxz. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find an equation of variation if a varies jointly as b and c, and a = 48 when b = 4 and c = 2. Solution We have a = kbc, so The variation constant is 6. The equation of variation is a = 6bc. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Joint variation is one form of combined variation Joint variation is one form of combined variation. In general, when a variable varies directly and/or inversely, at the same time, with more than one other variable, there is combined variation. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find an equation of variation if y varies jointly as x and z and inversely as the product of w and p, and y = 60 when x = 24, z = 5, w = 2, and p = 3. Solution The equation if variation is of the form so, substituting, we have: Thus, Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley