Direct Variation Two types of mathematical models occur so often that they are given special names. The first is called direct variation and occurs when one quantity is a constant multiple of the other, so we use an equation of the form y = kx to model this dependence.
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
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.
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.
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
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
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 2. The constant k = 1080 is the approximate speed of sound (in ft/s). Figure 2
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.
Inverse Variation
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
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 0.106 m3 at 25C is 50 kPa. Find the constant of proportionality, and write the equation that expresses the inverse proportionality. (b) If the sample expands to a volume of 0.3 m3, find the new pressure.
Example 2 – Solution (a) 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 = 0.106.
Example 2 – Solution Substituting these values in the equation, we get 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
Example 2 – Solution (b) When V = 0.3, we have cont’d (b) When V = 0.3, we have So the new pressure is about 17.7 kPa.
Joint 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.
Joint Variation For example, if we say that z is proportional to x and inversely proportional to y.
Example 3 – Newton’s Law of Gravitation Newton’s Law of Gravitation says that two objects with masses m1 and m2 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 Newton’s Law of Gravitation as an equation. Solution: Using the definitions of joint and inverse variation and the traditional notation G for the gravitational constant of proportionality, we have