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Direct Variation. 3.7: Modeling Using Variation. Certain formulas occur so frequently in applied situations that they are given special names. Variation formulas show how one quantity changes in relation to other quantities. Quantities can vary directly, inversely, or jointly.

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direct variation
Direct Variation

3.7: Modeling Using Variation

Certain formulas occur so frequently in applied situations that they are given special names. Variation formulas show how one quantity changes in relation to other quantities. Quantities can vary directly, inversely, or jointly.

Direct Variation

If a situation is described by an equation in the form

y = kx

where k is a constant, we say that yvaries directly as x. The number k is called the constant of variation.

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3.7: Modeling Using Variation

Direct Variation

Our work up to this point provides a step-by-step procedure for solving variation problems. This procedure applies to direct variation problems as well as to the other kinds of variation problems that we will discuss.

Solving Variation Problems

1. Write an equation that describes the given English statement.

2.Substitute the given pair of values into the equation in step 1 and find the value ofk.

3. Substitute the value of kinto the equation in step 1.

4. Use the equation from step 3 to answer the problem's question.

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Direct Variation

3.7: Modeling Using Variation

Direct variation with powers is modeled by polynomial functions.

Direct Variation with Powers

yvaries directly as the nth power of x if there exists some nonzero constant k such that

y = kxn.

example solving a direct variation problem
EXAMPLE: Solving a Direct Variation Problem

Solution

Step 1 Write an equation. We know that y varies directly as x is expressed as

y = kx.

By changing letters, we can write an equation that describes the following English statement: Garbage production, G, varies directly as the population, P.

G=kP

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3.7: Modeling Using Variation

The amount of garbage, G, varies directly as the population, P. Allegheny County, Pennsylvania, has a population of 1.3 million and creates 26 million pounds of garbage each week. Find the weekly garbage produced by New York City with a population of 7.3 million.

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EXAMPLE: Solving a Direct Variation Problem

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3.7: Modeling Using Variation

The amount of garbage, G, varies directly as the population, P. Allegheny County, Pennsylvania, has a population of 1.3 million and creates 26 million pounds of garbage each week. Find the weekly garbage produced by New York City with a population of 7.3 million.

Solution

Step 2 Use the given values to find k. Allegheny County has a population of 1.3 million and creates 26 million pounds of garbage weekly. Substitute 26 for G and 1.3 for P in the direct variation equation. Then solve for k.

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EXAMPLE: Solving a Direct Variation Problem

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3.7: Modeling Using Variation

The amount of garbage, G, varies directly as the population, P. Allegheny County, Pennsylvania, has a population of 1.3 million and creates 26 million pounds of garbage each week. Find the weekly garbage produced by New York City with a population of 7.3 million.

Solution

Step 3 Substitute the value of k into the equation.

G=kPUse the equation from step 1.

G= 20PReplace k, the constant of variation, with 20.

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EXAMPLE: Solving a Direct Variation Problem

3.7: Modeling Using Variation

The amount of garbage, G, varies directly as the population, P. Allegheny County, Pennsylvania, has a population of 1.3 million and creates 26 million pounds of garbage each week. Find the weekly garbage produced by New York City with a population of 7.3 million.

Solution

Step 4Answer the problem's question. New York City has a population of 7.3 million. To find its weekly garbage production, substitute 7.3 for P in

G= 20P and solve for G.

G = 20PUse the equation from step 3.

G = 20(7.3) Substitute 7.3 for P.

G = 146

The weekly garbage produced by New York City weighs approximately 146million pounds.

inverse variation
Inverse Variation

Inverse Variation

If a situation is described by an equation in the form

y=

where k is a constant, we say that yvaries inversely as x. The number k iscalled the constant of variation.

3.7: Modeling Using Variation

When two quantities vary inversely, one quantity increases as the other decreases, and vice versa. Generalizing, we obtain the following statement.

We use the same procedure to solve inverse variation problems as we did to solve direct variation problems.

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Solution

Step 1 Write an equation. We know that y varies inversely as x is expressed as

By changing letters, we can write an equation that describes the following English statement: The number of new songs each year, S, varies inversely as the number of years, N.

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3.7: Modeling Using Variation

EXAMPLE: Solving an Inverse Variation Problem

To continue making money, the number of new songs, S, a rock band needs to record each year varies inversely as the number of years, N, the band has been recording. After 4 years of recording, a band needs to record 15 new songs per year to be profitable. After 6 years, how many new songs will the band need to record in order to make a profit in the seventh year?

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3.7: Modeling Using Variation

EXAMPLE: Solving an Inverse Variation Problem

To continue making money, the number of new songs, S, a rock band needs to record each year varies inversely as the number of years, N, the band has been recording. After 4 years of recording, a band needs to record 15 new songs per year to be profitable. After 6 years, how many new songs will the band need to record in order to make a profit in the seventh year?

Solution

Step 2 Use the given values to find k. After 4 years of recording, the band needs to record 15 new songs. Substitute 15 for S and 4 for Nin the inverse variation equation. Then solve for k.

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3.7: Modeling Using Variation

EXAMPLE: Solving an Inverse Variation Problem

To continue making money, the number of new songs, S, a rock band needs to record each year varies inversely as the number of years, N, the band has been recording. After 4 years of recording, a band needs to record 15 new songs per year to be profitable. After 6 years, how many new songs will the band need to record in order to make a profit in the seventh year?

Solution

Step 3 Substitute the value of kinto the equation.

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3.7: Modeling Using Variation

EXAMPLE: Solving an Inverse Variation Problem

To continue making money, the number of new songs, S, a rock band needs to record each year varies inversely as the number of years, N, the band has been recording. After 4 years of recording, a band needs to record 15 new songs per year to be profitable. After 6 years, how many new songs will the band need to record in order to make a profit in the seventh year?

Solution

Step 4Answer the problem's question. We need to find how many new songs will the band need to record after 6 years in order to make a profit in the seventh year. Substitute 6 for N in the equation from step 3 and solve for S.

The band will need to record 10 new songs after 6 years.

joint variation
Joint Variation

3.7: Modeling Using Variation

Joint Variation

Joint variation is a variation in which a variable varies directly as the product of two or more other variables. Thus, the equation y=kxz is read "y varies jointly as x and z."

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3.7: Modeling Using Variation

EXAMPLE: Modeling Centrifugal Force

The centrifugal force, C, of a body moving in a circle varies jointly with the radius of the circular path, r, and the body's mass, m, and inversely with the square of the time, t, it takes to move about one full circle. A 6-gram body moving in a circle with radius 100 centimeters at a rate of 1 revolution in 2 seconds has a centrifugal force of 6000 dynes. Find the centrifugal force of an 18-gram body moving in a circle with radius 100 centimeters at a rate of 1 revolution in 3 seconds.

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Translate "Centrifugal force, C, varies jointly with radius, r, and mass, m, and inversely with the square of time, t."

If r= 100, m= 6, and t= 2, then C= 6000.

Solve for k.

Substitute 40 for k in the model for centrifugal force.

Find C when r= 100, m= 18, and t= 3.

3.7: Modeling Using Variation

EXAMPLE: Modeling Centrifugal Force

Solution

The centrifugal force is 8000 dynes.