Section 2.6 Variation and Applications Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
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Transcript of Section 2.6 Variation and Applications Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
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.
, 0y kx k
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 = kThe variation constant 14, is the rate of change of y with respect to x. The equation of variation is y = 14x.
42
3k
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 $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.
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, ).
, 0k
y kx
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.
220.4
(0.4)22
8.8
ky
xk
k
k
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.
( )
(12)12
kt P
Pk
t
18012
2160
k
k
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.
2160( )
2160(15)
15144
t PP
t
t
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.
ny kx
n
ky
x
Example
Find the equation of variation in which y varies directly as the square of x, and y = 12 when x = 2.
Thus y = 3x2.
2
212 2
12 4
3
y kx
k
k
k
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
2
2
3 20105
2
xzy k
w
k
105 15
7
k
k
2 2
77 or
xz xzy y
w w
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.
2
2
503
106
IE k
Dk
k
2
6 27
92
E
E