1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Variation Solve direct variation problems. Solve inverse variation problems. Solve joint and combined variation problems. SECTION

3 © 2010 Pearson Education, Inc. All rights reserved DIRECT VARIATION A quantity y is said to vary directly as the quantity x, or y is directly proportional to x, if there is a constant k such that y = kx. This constant k ≠ 0 is called the constant of variation or the constant of proportionality.

4 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving the Variation Problems OBJECTIVE Solve variation problems. Step 1 Write the equation with the constant of variation, k. EXAMPLE Suppose y varies as x and x = when y = 20. Find y when x = 8. y = kxy varies as x.

5 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving the Variation Problems OBJECTIVE Solve variation problems. Step 2 Substitute the given values of the variables into the equation in Step 1 to find the value of the constant k. Replace y with 20 and x with and solve for k. EXAMPLE Suppose y varies as x and x = when y = 20. Find y when x = 8.

6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving the Variation Problems OBJECTIVE Solve variation problems. Step 3 Rewrite the equation in Step 1 with the value of k from Step 2. y = 15x Replace k with 15. EXAMPLE Suppose y varies as x and x = when y = 20. Find y when x = 8.

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving the Variation Problems OBJECTIVE Solve variation problems. Step 4 Use the equation from Step 3 to answer the question posed in the problem. EXAMPLE Suppose y varies as x and x = when y = 20. Find y when x = 8. Replace x with 8 and simplify. So y = 120 when x = 8.

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Direct Variation in Electrical Circuits The current in a circuit connected to a 220-volt battery is 50 amperes. If the current in this circuit is directly proportional to the voltage of the attached battery, what voltage battery is needed to produce a current of 75 amperes? Solution Let I = current in amperes and V = voltage in volts of the battery. Step 1I = kV

9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Direct Variation in Electrical Circuits Solution continued Step 2 Step 3

10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Direct Variation in Electrical Circuits Solution continued Substitute I = 75 and solve for V.Step 4 A battery of 330 volts is needed to produce 75 amperes of current.

11 © 2010 Pearson Education, Inc. All rights reserved DIRECT VARIATION WITH POWERS “A quantity y varies directly as the nth power of x” means y = kx n where k is a nonzero constant and n > 0.

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Solving a Problem Involving Direct Variation with Powers Suppose that you had forgotten the formula for the volume of a sphere, but were told that the volume V of a sphere varies directly as the cube of its radius r. In addition, you are given that V = 972π when r = 9. Find V when r = 6. Solution Step 1V = kr 3

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Solution continued Step 2 Solving a Problem Involving Direct Variation with Powers

14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Solution continued Substitute r = 6 and solve for V.Step 4 Solving a Problem Involving Direct Variation with Powers Step 3

15 © 2010 Pearson Education, Inc. All rights reserved INVERSE VARIATION “A quantity y varies inversely as the quantity x,” or “y is inversely proportional to x” if where k is a nonzero constant.

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Solving an Inverse Variation Problem Suppose y varies inversely as x, and y = 35 when x = 11. Find y if x = 55. Solution Step 1 Step 2

17 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Solving an Inverse Variation Problem Solution continued Step 3 Step 4

18 © 2010 Pearson Education, Inc. All rights reserved “A quantity y varies inversely as the nth power of x” means that where k is a nonzero constant and n > 0. INVERSE VARIATION WITH POWERS

19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a Problem Involving Inverse Variation with Powers The intensity of light varies inversely as the square of the distance from the light source. If Rita doubles her distance from a lamp, what happens to the intensity of light at her new location?

20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a Problem Involving Inverse Variation with Powers Replace d by 2d, the new intensity I 1 is Solution Let I be the intensity of light at a distance d from the light source.

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Solving a Problem Involving Inverse Variation with Powers Solution continued If Rita doubles her distance from the lamp, the intensity is reduced to one-fourth of the intensity at the original location.

22 © 2010 Pearson Education, Inc. All rights reserved JOINT AND COMBINED VARIATION The expression “z varies jointly as x and y” means z = kxy for some nonzero constant k. If n and m are positive numbers, then the expression “z varies jointly as the nth power of x and the mth power of y” means z = kx n y m for some nonzero constant k.

23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Newton’s Law of Universal Gravitation Newton’s Law of Universal Gravitation: every object in the universe attracts every other object. This attracting force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the two objects. a.Write the law symbolically.

24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Newton’s Law of Universal Gravitation Solution Let m 1 and m 2 be the masses of the two objects and r be the distance between their centers.

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Newton’s Law of Universal Gravitation The constant of proportionality G is called the universal gravitational constant (approximately 6.67 × 10 –11 m 3 /kg/sec 2 ). F = the gravitational force between the objects. Solution continued

26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Newton’s Law of Universal Gravitation b.Estimate the value of g (the acceleration due to gravity) near the surface of the Earth. Use the estimates: radius of Earth R E = 6.38 × 10 6 m mass of the Earth M E = 5.98 × kg Solution

27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Newton’s Law of Universal Gravitation Solution continued