Transcript y - B Math 144
2.5 Variation and Applications
Find equations of direct, inverse, and combined variation given values of the variables.
Solve applied problems involving variation.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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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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.
y
kx
,
k
0
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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Direct Variation
Example
: Find the variation constant and an equation of variation in which when
x
= 3.
y
varies directly as
x
, and
y
= 42
Solution
: We know that (3, 42) is a solution of
y
=
kx
.
42
y
=
kx = k
3 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
= 14
x
.
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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.
W
W(
h
) =
kh
(18) =
k
$168.30 =
k
18 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.
W(33) = $9.35(33) = $308.55
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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Inverse Variation
If a situation gives rise to a function
f
(
x
) =
k
/
x
, or
y
number
k
is called the
variation constant, constant of proportionality
.
or =
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 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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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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, ).
y
k x
,
k
0
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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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
:
y
22 (0.4)22
k x k
0.4
k
8.8
k
The variation constant is 8.8. The equation of variation is
y
= 8.8/
x
. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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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.
k P t
(12) 180
k
12
k
12 2160
k t
varies inversely as
P
This is the variation constant.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 2.5 - 9
Application continued
The equation of variation is
t
(
P
) = 2160/
P.
Next we compute
t
(15).
2160
P t
(15)
t
2160 15 144 It would take 144 days for 15 people to complete the same job.
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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Combined Variation
Other kinds of variation:
y
varies
directly as the nth power of x
positive constant
k y
if there is some
n kx y
varies
inversely as the nth power of x
if there is some positive constant
k
such that .
x n y
varies
jointly as x and z
constant
k
such that
y
= if there is some positive
kxz
.
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Example
Example
: Find the equation of variation in which
y
directly as the square of
x
, and
y
= 12 when
x
= 2.
varies
Solution
:
y
kx
2 12 2 2 12 3
k k
Thus
y =
3
x
2 .
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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Example
: Find the equation of variation in which
y
jointly as
x
and
z
, and
y
= 42 when
x
= 2 and
z
= 3. varies
Solution
:
y
kxz
42 42 7
k k
Thus
y = 7xz
.
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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
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
Solution
:
y xz w
2 105 105 7
k
15 2 2 Thus
y
xz
7
w
2 or
y
7
xz
.
w
2 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 2.5 - 14
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.
E
3
I k k
D
2 50 10 2 Solve for
k
.
6
k
Substitute the second set of data into the equation.
E
6
I D
2 The lamp gives an luminance reading of 2 units.
E E
2 9 2
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