1. College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson

2. Polynomial and Rational Functions 3

3. Modeling Variation 3.8

4. Fundamentals When scientists talk about a mathematical model for a real-world phenomenon, they often mean an equation that describes the dependence of one physical quantity on another. For instance, the model may describe the population of an animal species as a function of time or the pressure of a gas as a function of its volume.

5. Fundamentals In this section, we study a kind of modeling called variation.

6. Direct Variation

7. One type of variation is called direct variation; it occurs when one quantity is a constant multiple of the other. We use a function of the form f(x) = kx to model this dependence.

8. Direct Variation If the quantities x and y are related by an equation y = kx for some constant k ≠ 0, we say that: y varies directly as x. y is directly proportional to x. y is proportional to x. The constant k is called the constant of proportionality.

9. Direct Variation Recall that the graph of an equation of the form y = mx + b is a line with: Slope m y-intercept b

10. Direct Variation So, the graph of an equation y = kx that describes direct variation is a line with: Slope k y-intercept 0

11. E.g. 1—Direct Variation During a thunderstorm, you see the lightning before you hear the thunder because light travels much faster than sound. The distance between you and the storm varies directly as the time interval between the lightning and the thunder.

12. E.g. 1—Direct Variation (a)Suppose that the thunder from a storm 5,400 ft away takes 5 s to reach you. Determine the constant of proportionality and write the equation for the variation.

13. E.g. 1—Direct Variation (b) Sketch the graph of this equation. What does the constant of proportionality represent? (c) If the time interval between the lightning and thunder is now 8 s, how far away is the storm?

14. E.g. 1—Direct Variation Let d be the distance from you to the storm and let t be the length of the time interval. We are given that d varies directly as t. So, d = kt where k is a constant. Example (a)

15. E.g. 1—Direct Variation To find k, we use the fact that t = 5 when d = 5400. Substituting these values in the equation, we get: 5400 = k(5) Example (a)

16. E.g. 1—Direct Variation Substituting this value of k in the equation for d, we obtain: d = 1080t as the equation for d as a function of t. Example (a)

17. E.g. 1—Direct Variation The graph of the equation d = 1080t is a line through the origin with slope 1080. The constant k = 1080 is the approximate speed of sound (in ft/s). Example (b)

18. E.g. 1—Direct Variation When t = 8, we have: d = 1080 ∙ 8 = 8640 So, the storm is 8640 ft ≈ 1.6 mi away. Example (c)

19. Inverse Variation

20. Another function that is frequently used in mathematical modeling is where k is a constant.

21. Inverse Variation If the quantities x and y are related by the equation for some constant k ≠ 0, we say that: y is inversely proportional to x. y varies inversely as x.

22. Inverse Variation The graph of y = k/x for x > 0 is shown for the case k > 0. It gives a picture of what happens when y is inversely proportional to x.

23. E.g. 2—Inverse Variation Boyle’s Law states that: When a sample of gas is compressed at a constant temperature, the pressure of the gas is inversely proportional to the volume of the gas.

24. E.g. 2—Inverse Variation (a)Suppose the pressure of a sample of air that occupies 0.106 m 3 at 25°C is 50 kPa. Find the constant of proportionality. Write the equation that expresses the inverse proportionality.

25. E.g. 2—Inverse Variation (b)If the sample expands to a volume of 0.3 m 3, find the new pressure.

26. E.g. 2—Inverse Variation Let P be the pressure of the sample of gas and let V be its volume. Then, by the definition of inverse proportionality, we have: where k is a constant. Example (a)

27. E.g. 2—Inverse Variation To find k, we use the fact that P = 50 when V = 0.106. Substituting these values in the equation, we get: k = (50)(0.106) = 5.3 Example (a)

28. E.g. 2—Inverse Variation Putting this value of k in the equation for P, we have: Example (a)

29. E.g. 2—Inverse Variation When V = 0.3, we have: So, the new pressure is about 17.7 kPa. Example (b)

30. Combining Different Types of Variation

31. In the sciences, relationships between three or more variables are common, and any combination of the different types of proportionality that we have discussed is possible.

32. Combining Different Types of Variation If the quantities x, y, and z are related by the equation z = kxy where k is a nonzero constant, we say that: z varies jointly as x and y. z is jointly proportional to x and y.

33. Combining Different Types of Variation In the sciences, relationships between three or more variables are common. Any combination of the different types of proportionality that we have discussed is possible. For example, if we say that z is proportional to x and inversely proportional to y.

34. E.g. 4—Newton’s Law of Gravitation Newton’s Law of Gravitation says that: Two objects with masses m 1 and m 2 attract each other with a force F that is jointly proportional to their masses and inversely proportional to the square of the distance r between the objects. Express the law as an equation.

35. E.g. 4—Newton’s Law of Gravitation Using the definitions of joint and inverse variation, and the traditional notation G for the gravitational constant of proportionality, we have:

36. Gravitational Force If m 1 and m 2 are fixed masses, then the gravitational force between them is: F = C/r 2 where C = Gm 1 m 2 is a constant.

37. Gravitational Force The figure shows the graph of this equation for r > 0 with C = 1. Observe how the gravitational attraction decreases with increasing distance.