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Inverse Variation

If a quantity, y "varies inversely as" x, then as x increases, y decreases and as x decreases, y increases.

The phrase, "y varies inversely as x", has equation form:

.xk

y Here, k is a positive constant called the constant of proportionality.

The phrase, "y varies inversely as x", is commonly written as "y is directly proportional to x".

The phrase, "y varies inversely as the square of x", has

.2x

ky equation form:

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Inverse Variation

Slide 2

The phrase, "y varies inversely as the cube of x", has

.3x

ky equation form:

square root of x", has equation form: .x

ky

The phrase, "y varies inversely as

Example 1: The diameter of a pulley varies inversely as the rate at which it spins (see figure).

If the larger pulley has a diameter of 2.5 inches and spins at a rate of 20 revolutions per minute, (rpm), find the rate at which the smaller pulley is spinning if it has a diameter of 1.5 inches.

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Inverse Variation

Slide 3

Let, y represent the rate at which a pulley spins and x represent its diameter. (It does not matter which is x and which is y.) Then write the algebraic equation for this inverse relationship.

xk

y

Next, find k by substituting the information given about the larger pulley.

.5.2

20k

50 = kNext, rewrite the inverse relationship equation with this value of k.

xy

50Next, use this equation to answer the question by substituting the diameter of the smaller pulley. 5.1

50y 33.3 rpm

The smaller pulley revolves at a rate of 33.3 rpm.

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Inverse Variation

Slide 4

Example 2: Determine if the data in the table can be

,nx

ky modeled by an equation of the form, where n is a

natural number. x y

2 400 4 50 8 6.75 10 3.2

One way to approach this is to by trying values for n. For example, let n = 1, so

.xk

y Then find the value of k by

substituting any pair of x and y-values from the table:

,2

400k k = 200, .

200x

y so

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Inverse Variation

Slide 5

x y

2 400 4 50 8 6.75 10 3.2

,200

xy However, the equation, will not

model all the other x, y-pairs from the table.

For example, x = 10 and y = 3.2, will not fit the model.

The next try would be to let n = 2. If this does not work then, successive trials would be n = 3, n = 4, etc.Try this approach and see which, if any, work. The answer will be revealed on the next left mouse click.The data can be modeled with n = 3. NOTE: This can be solved later in a direct way using exponential functions.

Table of Contents

Inverse Variation