Section 2.6 Variation and Applications Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives  Find equations of direct, inverse, and combined variation given values of the variables.  Solve applied problems involving variation.

Direct Variation If a situation gives rise to a linear function f(x) = kx, or y = kx, where k is a positive constant, we say that we have direct variation, or that y varies directly as x, or that y is directly proportional to x. The number k is called the variation constant, or constant of proportionality.

Direct Variation The graph of y = kx, k > 0, always goes through the origin and rises from left to right. As x increases, y increases; that is, the function is increasing on the interval (0,  ). The constant k is also the slope of the line.

Example Find the variation constant and an equation of variation in which y varies directly as x, and y = 42 when x = 3. We know that (3, 42) is a solution of y = kx. y = kx 42 = k  3 14 = k The variation constant 14, is the rate of change of y with respect to x. The equation of variation is y = 14x.

Example Wages. A cashier earns an hourly wage. If the cashier worked 18 hours and earned $168.30, how much will the cashier earn if she works 33 hours? We can express the amount of money earned as a function of the amount of hours worked. I(h) = kh I(18) = k  18 $ = k  18 $9.35 = k The hourly wage is the variation constant. Next, we use the equation to find how much the cashier will earn if she works 33 hours. I(33) = $9.35(33) = $308.55

Inverse Variation If a situation gives rise to a function f(x) = k/x, or y = k/x, where k is a positive constant, we say that we have inverse variation, or that 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. For the graph y = k/x, k  0, as x increases, y decreases; that is, the function is decreasing on the interval (0,  ).

Inverse Variation For the graph y = k/x, k  0, as x increases, y decreases; that is, the function is decreasing on the interval (0,  ).

Example Find the variation constant and an equation of variation in which y varies inversely as x, and y = 22 when x = 0.4. The variation constant is 8.8. The equation of variation is y = 8.8/x.

Example Road Construction. The time t required to do a job varies inversely as the number of people P who work on the job (assuming that they all work at the same rate). If it takes 180 days for 12 workers to complete a job, how long will it take 15 workers to complete the same job? We can express the amount of time required, in days, as a function of the number of people working.

Example continued The equation of variation is t(P) = 2160/P. Next we compute t(15). It would take 144 days for 15 people to complete the same job.

Combined Variation Other kinds of variation: y varies directly as the nth power of x if there is some positive constant k such that. y varies inversely as the nth power of x if there is some positive constant k such that. y varies jointly as x and z if there is some positive constant k such that y = kxz.

Example Find the equation of variation in which y varies directly as the square of x, and y = 12 when x = 2. Thus y = 3x 2.

Example Find the equation of variation in which y varies jointly as x and z and inversely as the square of w, and y = 105 when x = 3, z = 20, and w = 2. Thus

Example The luminance of a light (E) varies directly with the intensity (I) of the light and inversely with the square distance (D) from the light. At a distance of 10 feet, a light meter reads 3 units for a 50-cd lamp. Find the luminance of a 27-cd lamp at a distance of 9 feet. Substitute the second set of data into the equation. The lamp gives an luminance reading of 2 units.