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Fundamentals
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2. Copyright © Cengage Learning. All rights reserved.
1.12
Modeling Variation
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3. Objectives Direct Variation Inverse Variation
Combining Different Types of Variation

4. Direct Variation

5. Direct Variation
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.

6. Direct Variation
We know that the graph of an equation of the form y = mx + b is a line with slope m and y-intercept b.
So the graph of an equation y = kx that describes direct variation is a line with slope k and y-intercept 0 (see Figure 1).
Figure 1

7. Example 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.
(a) Suppose that the thunder from a storm 5400 ft away takes 5 s to reach you. Determine the constant of proportionality, and write the equation for the variation.

8. Example 1 – Direct Variation
cont’d
(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?
Solution:
(a) Let d be the distance from you to the storm, and let t be the length of the time interval.

9. Example 1 – Solution We are given that d varies directly as t, so
cont’d
We are given that d varies directly as t, so
d = kt
where k is a constant. To find k, we use the fact that t = 5 when d = 5400.
Substituting these values in the equation, we get
5400 = k (5)
Substitute

10. Example 1 – Solution
cont’d
Substituting this value of k in the equation for d, we obtain
d = 1080t
as the equation for d as a function of t.
Solve for k

11. Example 1 – Solution
cont’d
(b) The graph of the equation d = 1080t is a line through the origin with slope 1080 and is shown in Figure The constant k = 1080 is the approximate speed of sound (in ft/s).
Figure 2

12. Example 1 – Solution (c) When t = 8, we have d = 1080  8 = 8640
cont’d
(c) When t = 8, we have
d = 1080  8 = 8640
So the storm is 8640 ft  1.6 mi away.

13. Inverse Variation

14. Inverse Variation

15. Inverse Variation
The graph of y = k/x for x > 0 is shown in Figure 3 for the case k > 0.
It gives a picture of what happens when y is inversely proportional to x.
Inverse variation
Figure 3

16. Example 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.
(a) Suppose the pressure of a sample of air that occupies m3 at 25C is 50 kPa. Find the constant of proportionality, and write the equation that expresses the inverse proportionality. Sketch a graph of this equation.
(b) If the sample expands to a volume of 0.3 m3, find the new pressure.

17. Example 2(a) – Solution
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. To find k, we use the fact that P = 50 when V =

18. Example 2(a) – Solution
cont’d
Substituting these values in the equation, we get
k = (50)(0.106) = 5.3
Putting this value of k in the equation for P, we have
Substitute
Solve for k

19. Example 2(a) – Solution The graph is shown in Figure 4. cont’d

20. Example 2(b) – Solution When V = 0.3, we have
cont’d
When V = 0.3, we have
So the new pressure is about 17.7 kPa.

21. Combining Different Types of Variation

22. Combining Different Types of Variation
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.
For example, if the quantities x, y, and z are related by the equation
z = kxy
then we say that z is proportional to the product of x and y.

23. Combining Different Types of Variation
We can also express this relationship by saying that z varies jointly as x and y or that z is jointly proportional to x and y. If the quantities x, y, and z are related by the equation
we say that z is proportional to x and inversely proportional to y or that z varies directly as x and inversely as y.

24. Example 3 – Combining Variations
The apparent brightness B of a light source (measured in W/m2) is directly proportional to the luminosity L (measured in W) of the light source and inversely proportional to the square of the distance d from the light source (measured in meters).
(a) Write an equation that expresses this variation.
(b) If the distance is doubled, by what factor will the brightness change?
(c) If the distance is cut in half and the luminosity is tripled, by what factor will the brightness change?

25. Example 3 – Solution
(a) Since B is directly proportional to L and inversely proportional to d2, we have
where k is a constant.
(b) To obtain the brightness at double the distance, we replace d by 2d in the equation we obtained in part (a).
Brightness at distance d and luminosity L
Brightness at distance 2d

26. Example 3 – Solution
cont’d
Comparing this expression with that obtained in part (a), we see that the brightness is of the original brightness.
(c) To obtain the brightness at half the distance d and triple the luminosity L, we replace d by d/2 and L by 3L in the equation we obtained in part (a).
Comparing this expression with that obtained in part (a), we see that the brightness is 12 times the original brightness.
Brightness at distance
and luminosity 3L