September 26, S2.5 Variation and Applications

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1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3 The Composition of Functions 2.4 Symmetry and Transformations 2.5 Variation and Applications 2.5 Variation and Applications Find equations of direct, inverse, and combined variation given values of the variables. Solve applied problems involving 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. 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. 1

2 Example: Find the variation constant and an equation of variation in which y varies directly as x, and y = 42 when x = 3. Solution: 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. Application 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? Solution: 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) = $ 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, ). For the graph y = k/x, k > 0, as x increases, y decreases; that is, the function is decreasing on the interval (0, ). 2

3 Example: Find the variation constant and an equation of variation in which y varies inversely as x, and y = 22 when x = 0.4. Solution: Application 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? Solution: We can express the amount of time required, in days, as a function of the number of people working. t varies inversely as P The variation constant is 8.8. The equation of variation is y = 8.8/x. This is the variation constant. Application continued The equation of variation is t(p) = 2160/P. Next we compute t(15). 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. It would take 144 days for 15 people to complete the same job. y varies jointly as x and z if there is some positive constant k such that y = kxz. 3

4 Find the variation constant and an equation of variation for the given situation. 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. 224/2. y varies directly as x, and y = 0.1 when x = 0.2. Solve for k. Substitute the second set of data into the equation with the k value. The lamp gives an luminance reading of 2 units. 225/16. Rate of Travel. The time t required to drive a fixed distance varies inversely as the speed r. It takes 5 hr at a speed of 80 km/h to drive a fixed distance. How long will it take to drive the same distance at a speed of 70 km/h? 225/26. Find an equation of variation for the given situation. y varies inversely as the square of x, and y = 6 when x = 3. 4

5 225/34. Find an equation of variation for the given situation. y varies jointly as x and z and inversely as the square of w, and y = 12/5, when x = 16, z = 3, and w = /40. Boyles Law. The volume V of a given mass of a gas varies directly as the temperature T and inversely as the pressure P. If V = 231 cm 3 when T = 42 o and P = 20 kg/cm 2, what is the volume when T = 30 o and P = 15 kg/cm 2? 5