Fundamentals
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Making
Models
Using
1.11
Variation
Copyright © Cengage Learning. All rights reserved.
Objectives
► Direct Variation
► Inverse Variation
► Joint Variation
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Direct Variation
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Direct Variation
Two types of mathematical models occur so often that they
are given special names.
The first is called direct variation and occurs when one
quantity is a constant multiple of the other, so we use an
equation of the form y = kx to model this dependence.
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Direct Variation
We know that the graph of an equation of the form
y = mx + b is a line with slope m and y-intercept b.
So the graph of an equation y = kx that describes direct
variation is a line with slope k and y-intercept 0
(see Figure 1).
Figure 1
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Example 1 – Direct Variation
During a thunderstorm you see the lightning before you
hear the thunder because light travels much faster than
sound.
The distance between you and the storm varies directly
as the time interval between the lightning and the
thunder.
(a) Suppose that the thunder from a storm 5400 ft away
takes 5 s to reach you. Determine the constant of
proportionality, and write the equation for the variation.
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Example 1 – Direct Variation
cont’d
(b) Sketch the graph of this equation. What does the
constant of proportionality represent?
(c) If the time interval between the lightning and thunder is
now 8 s, how far away is the storm?
Solution:
(a) Let d be the distance from you to the storm, and let t be
the length of the time interval.
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Example 1 – Solution
cont’d
We are given that d varies directly as t, so
d = kt
where k is a constant. To find k, we use the fact that
t = 5 when d = 5400.
Substituting these values in the equation, we get
5400 = k(5)
Substitute
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Example 1 – Solution
cont’d
Solve for k
Substituting this value of k in the equation for d, we
obtain
d = 1080t
as the equation for d as a function of t.
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Example 1 – Solution
cont’d
(b) The graph of the equation d = 1080t is a line through
the origin with slope 1080 and is shown in Figure 2.
The constant k = 1080 is the approximate speed of
sound (in ft/s).
Figure 2
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Example 1 – Solution
cont’d
(c) When t = 8, we have
d = 1080  8 = 8640
So the storm is 8640 ft  1.6 mi away.
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Inverse Variation
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Inverse Variation
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Inverse Variation
The graph of y = k/x for x > 0 is shown in Figure 3 for the
case k > 0.
It gives a picture of what happens when y is inversely
proportional to x.
Inverse variation
Figure 3
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Example 2 – Inverse Variation
Boyle’s Law states that when a sample of gas is
compressed at a constant temperature, the pressure of the
gas is inversely proportional to the volume of the gas.
(a) Suppose the pressure of a sample of air that occupies
0.106 m3 at 25C is 50 kPa.
Find the constant of proportionality, and write the
equation that expresses the inverse proportionality.
(b) If the sample expands to a volume of 0.3 m3, find the
new pressure.
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Example 2 – Solution
(a) Let P be the pressure of the sample of gas, and let V
be its volume.
Then, by the definition of inverse proportionality,
we have
where k is a constant. To find k, we use the fact that
P = 50 when V = 0.106.
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Example 2 – Solution
cont’d
Substituting these values in the equation, we get
Substitute
k = (50)(0.106) = 5.3
Solve for k
Putting this value of k in the equation for P, we have
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Example 2 – Solution
cont’d
(b) When V = 0.3, we have
So the new pressure is about 17.7 kPa.
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Joint Variation
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Joint Variation
In the sciences, relationships between three or more
variables are common, and any combination of the different
types of proportionality that we have discussed is possible.
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Joint Variation
For example, if
we say that z is proportional to x and inversely
proportional to y.
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Example 3 – Newton’s Law of Gravitation
Newton’s Law of Gravitation says that two objects with
masses m1 and m2 attract each other with a force F that is
jointly proportional to their masses and inversely
proportional to the square of the distance r between the
objects.
Express Newton’s Law of Gravitation as an equation.
Solution:
Using the definitions of joint and inverse variation and the
traditional notation G for the gravitational constant of
proportionality, we have
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