Transcript Beginning & Intermediate Algebra, 4ed

§ 8.4
Variation and Problem
Solving
Direct Variation
y varies directly as x, or y is directly proportional to
x, if there is a nonzero constant k such that y = kx.
The family of equations of the form y = kx are
referred to as direct variation equations.
The number k is called the constant of variation or
the constant of proportionality.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Direct Variation
If y varies directly as x, find the constant of
variation k and the direct variation equation,
given that y = 5 when x = 30.
y = kx
5 = k·30
k = 1/6
1
So the direct variation equation is y =
x.
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Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Direct Variation
Example:
If y varies directly as x, and y = 48 when x = 6, then
find y when x = 15.
y = kx
48 = k·6
8=k
So the equation is y = 8x.
y = 8·15
y = 120
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Direct Variation
Example:
At sea, the distance to the horizon is directly
proportional to the square root of the elevation
of the observer. If a person who is 36 feet
above water can see 7.4 miles, find how far a
person 64 feet above the water can see.
Round your answer to two decimal places.
Continued.
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Direct Variation
Example continued:
d k e
7.4  k 36
7.4  6 k
7.4
k
6
So our equation is
We substitute our given value
for the elevation into the
equation.
7 .4
d
64
6
7.4
59.2
d
(8) 
 9.87 miles
6
6
7.4
d
e
6
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Inverse Variation
y varies inversely as x, or y is inversely proportional
to x, if there is a nonzero constant k such that y = k/x.
The family of equations of the form y = k/x are
referred to as inverse variation equations.
The number k is still called the constant of variation
or the constant of proportionality.
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Inverse Variation
Example:
If y varies inversely as x, find the constant of
variation k and the inverse variation equation,
given that y = 63 when x = 3.
y = k/x
63 = k/3
63·3 = k
189 = k
189
So the inverse variation equation is y =
x
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Powers of x
y can vary directly or inversely as powers of x,
as well.
y varies directly as a power of x if there is a
nonzero constant k and a natural number n
such that y = kxn.
y varies inversely as a power of x if there is a
nonzero constant k and a natural number n
such that y = k .
x
n
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Powers of x
Example:
The maximum weight that a circular column
can hold is inversely proportional to the
square of its height.
If an 8-foot column can hold 2 tons, find how
much weight a 10-foot column can hold.
Continued.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Powers of x
Example continued:
k
w 2
h
k
k
2 2 
8
64
We substitute our given value
for the height of the column
into the equation.
128 128
w 2 
 1.28 tons
10
100
k  128
So our equation is
128
w 2
h
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Variation and Problem Solving
Example:
Kathy spends 1.5 hours watching television and 8 hours studying
each week. If the amount of time spent watching TV varies inversely
with the amount of time spent studying, find the amount of time
Kathy will spend watching TV if she studies 14 hours a week.
1.) Understand
Read and reread the problem.
2.) Translate
We are told that the amount of time watching TV varies inversely
with the amount of time spent studying.
Let T = the number of hours spent watching television.
Let s = the number of hours spent studying.
k
T
s
Continued
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Variation and Problem Solving
Example continued:
3.) Solve
To find k, substitute T = 1.5 and s = 8.
k
1.5 
8
12  k
We now write the variation equation with k replaced by 12.
T
12
s
Replace s by 14 and find the value of T.
12
14
 0.86
T
Martin-Gay, Beginning and Intermediate Algebra, 4ed
Continued
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Variation and Problem Solving
Example continued:
3.) Interpret
Kathy will spend approximately 0.86 hours (or 52 minutes)
watching TV.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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