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Rational Expressions,
Equations, and Functions
Rational Expressions and Functions:
Multiplying and Dividing
Rational Expressions and Functions:
Adding and Subtracting
Complex Rational Expressions
Rational Equations
Solving Applications Using Rational
Equations
Division of Polynomials
Synthetic Division
Formulas, Applications, and Variation
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
6.8
Formulas, Applications,
and Variation

Formulas

Direct Variation

Inverse Variation

Joint and Combined Variation
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Formulas
Formulas occur frequently as mathematical
models. Many formulas contain rational
expressions, and to solve such formulas for a
specified letter, we proceed as when solving
rational equations.
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Slide 6- 4
Example
In a hydraulic system, a fluid is confined to two
connecting chambers. The pressure in each
chamber is the same and is given by finding the
force exerted (F) divided by the surface area (A).
Therefore, we know
F1 F2
 .
A1 A2
Solve for A2.
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Slide 6- 5
Solution
F1
F2
A1 A2   A1 A2 
A1
A2
A2 F1  A1F2
A1F2
A2 
F1
Multiplying both sides by the LCD
Simplifying by removing factors
Dividing both sides by F1
This formula can be used to calculate A2 whenever
A1, F2, and F1 are known.
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To Solve a Rational Equation for a
Specified Variable
1. If necessary, multiply both sides by the LCD to
clear fractions.
2. Multiply, as needed, to remove parentheses.
3. Get all terms with the specified variable alone on
one side.
4. Factor out the specified variable if it is in no more
than one term.
5. Multiply or divide on both sides to isolate the
specified variable.
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Slide 6- 7
Variation
To extend our study of formulas and
functions, we now examine three real-world
situations: direct variation, inverse variation,
and combined variation.
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Slide 6- 8
Direct Variation
A mass transit driver earns $17 per hour. In 1 hr, $17
is earned. In 2 hr, $34 is earned. In 3 hr, $51 is
earned, and so on. This gives rise to a set of ordered
pairs: (1, 22), (2, 34), (3, 51), (4, 68), and so on.
Note that the ratio of earnings E to time t is 17/1 in
every case.
If a situation gives rise to pairs of numbers in which
the ratio is constant, we say that there is direct
variation. Here earnings vary directly as the time:
We have E/t = 17, so E = 17t, or using function
notation, E(t) = 17t.
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Slide 6- 9
Direct Variation
When a situation gives rise to a linear
function of the form f(x) = kx, or y = kx,
where k is a nonzero constant, we say that
there is direct variation, that y varies directly
as x, or that y is proportional to x. The
number k is called the variation constant, or
constant of proportionality.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 10
Example
Find the variation constant and an
equation of variation if y varies directly as x, and
y = 15 when x = 3.
Solution
We know that (3, 15) is a solution of y = kx.
Therefore,
Substituting
15  k  3
15
 k ; or k  5
3
Solving for k
The variation constant is 5. The equation of variation
is y = 5x. The notation y(x) = 5x or f(x) = 5x is also
used.
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Inverse Variation
To see what we mean by inverse variation,
suppose it takes one person 8 hours to paint
the baseball fields for the local park district. If
two people do the job, it will take only 4
hours. If three people paint the fields, it will
take only 2 and 2/3 hours, and so on. This
gives rise to pairs of numbers, all have the
same product: (1, 8), (2, 4), (3, 8/3), (4, 2),
and so on.
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Inverse Variation
Note that the product of each pair of numbers
is 8. Whenever a situation gives rise to pairs
of numbers for which the product is constant,
we say that there is inverse variation. Since
pt = 8, the time t, in hours, required for the
fields to be painted by p people is given by
t = 8/p or, using function notation, t(p) = 8/p.
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Slide 6- 13
Inverse Variation
When a situation gives rise to a rational
function of the form f(x) = k/x, or y = k/x,
where k is a nonzero constant, we say that
there is inverse variation, that is 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.
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Slide 6- 14
Example
The time, t, required to empty a tank varies
inversely as the rate, r, of pumping. If a pump can
empty a tank in 90 minutes at the rate of 1080
kL/min, how long will it take the pump to empty
the same tank at the rate of 1500 kL/min?
Solution
1. Familiarize. Because of the phrase
“ . . . varies inversely as the rate, r, of pumping,”
we express the amount of time needed to empty
the tank as a function of the rate: t(r) = k/r
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Slide 6- 15
Solution (continued)
2. Translate. We use the given information to
solve for k. Then we use that result to write the
equation of variation.
k
t (r ) 
r
Using function notation
k
Replacing r with 1080
1080
k
Replacing t(90) with 1080
90 
1080
Solving for k, the variation constant
97, 200  k
t (1080) 
The equation of variation is t(r) = 97,200/r. This
is the translation.
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Slide 6- 16
Solution (continued)
3. Carry out. To find out how long it would
take to pump out the tank at a rate of 1500
mL/min, we calculate t(1500).
97, 200
t (1500) 
 64.8.
1500
t = 64.8 when r = 1500
4. Check. We could now recheck each step.
Note that, as expected, as the rate goes up, the
time it takes goes down.
5. State. If the pump is emptying the tank at a
rate of 1500 mL/min, then it will take 64.8
minutes to empty the entire tank.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 17
Joint and Combined Variation
When a variable varies directly with more
than one other variable, we say that there is
joint variation. For example, in the formula
for the volume of a right circular cylinder,
V = πr2h, we say that V varies jointly as h
and the square of r.
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Joint Variation
y varies jointly as x and z if, for some
nonzero constant k, y = kxz.
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Example
Find an equation of variation if a
varies jointly as b and c, and a = 48 when b =
4 and c = 2.
Solution
We have a = kbc, so
48  k  4  2
6k
The variation constant is 6.
The equation of variation is a = 6bc.
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Joint variation is one form of combined
variation. In general, when a variable varies
directly and/or inversely, at the same time,
with more than one other variable, there is
combined variation.
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Slide 6- 21
Example
Find an equation of variation if y varies
jointly as x and z and inversely as the product of w and
p, and y = 60 when x = 24, z = 5, w = 2, and p = 3.
Solution The equation if variation is of the form
xz
y k
wp
so, substituting, we have:
24  5
60  k 
23
60  k  20
3k
xz
Thus, y  3 
.
wp
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