171S2.5 Variation and Applications
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
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.
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2.5 Variation and Applications
• Find equations of direct, inverse, and combined variation given values of the variables.
• Solve applied problems involving variation.
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.
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171S2.5 Variation and Applications
Direct Variation
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.
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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
$168.30 = 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.
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, ∞).
For the graph y = k/x, k > 0, as x increases, y decreases; that is, the function is decreasing on the interval (0, ∞).
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171S2.5 Variation and Applications
Inverse Variation
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:
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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.
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171S2.5 Variation and Applications
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 50­cd lamp. Find the luminance of a 27­cd lamp at a distance of 9 feet.
Solve for k.
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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 hm/h?
Substitute the second set of data into the equation with the k value.
The lamp gives an luminance reading of 2 units.
Find the variation constant and an equation of variation for the given situation.
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.
224/2. y varies directly as x, and y = 0.1 when x = 0.2.
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171S2.5 Variation and Applications
226/40. 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 cm3 when T = 42o and P = 20 kg/cm2, what is the volume when T = 30o and P = 15 kg/cm2? February 15, 2011
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 = 5.
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