Sect. 2.6
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.
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.
y=kx, k>0
Ex. 1
Find the variation constant and an equation of variation in which y varies directly as x, and y = 42 when x = 3.
Ex. 2
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? Ex. 3
The number of centimeters of water W produced from melting snow varies directly as S, the number of centimeters of snow. Meteorologists have found that under certain conditions 150cm of snow will melt to 16.8cm of water. to how many centimeters of water will 200cm of snow melt under the same conditions?
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, ∞).
Ex. 4
Find the variation constant and an equation of variation in which y varies inversely as x, and y = 22 when x = 0.4.
Ex. 5
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?
Ex. 6
The time t required to fill a swimming pool varies inversely as the rate of flow r of water into the pool. A tank truck can fill a pool in 90 min. at a rate of 1500 L/min. How long would it take to fill the pool at a rate of 1800 L/min.?
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 = kxn. 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.
Ex. 7
Find the equation of variation in which y varies directly as the square of x, and y = 12 when x = 2. Ex. 8
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. Ex. 9
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 50­cd lamp. Find the luminance of a 27­cd lamp at a distance of 9 feet.