1.11

Making Models Using Variation

2

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.

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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

(b) Sketch the graph of this equation. What does the constant of proportionality represent?

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Example 1 – Direct Variation

(c) If the time interval between the lightning and thunder is now 8 s, how far away is the storm?

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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 the graph for the case k > 0.

It gives a picture of what happens when y is inversely proportional to x.

Inverse variation

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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 25C is 50 kPa.

Find the constant of proportionality, and write the equation that expresses the inverse proportionality.

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Example 2 – Inverse Variation

(b) If the sample expands to a volume of 0.3 m3, find the new pressure.

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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.

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Making Models Using Variation

Practice:

p. 121-123

#1-4, 5-17o, 22, 27-31o, 35