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A quantity y varies inversely with the square of x . If y = 8 when x = 3 , find y when x is 4.

9 2

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Solving problems involving joint variation

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation    . For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable c , cost, varies jointly with the number of students, n , and the distance, d .

Joint variation

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, if x varies directly with both y and z , we have x = k y z . If x varies directly with y and inversely with z , we have x = k y z . Notice that we only use one constant in a joint variation equation.

Solving problems involving joint variation

A quantity x varies directly with the square of y and inversely with the cube root of z . If x = 6 when y = 2 and z = 8 , find x when y = 1 and z = 27.

Begin by writing an equation to show the relationship between the variables.

x = k y 2 z 3

Substitute x = 6 , y = 2 , and z = 8 to find the value of the constant k .

6 = k 2 2 8 3 6 = 4 k 2 3 = k

Now we can substitute the value of the constant into the equation for the relationship.

x = 3 y 2 z 3

To find x when y = 1 and z = 27 , we will substitute values for y and z into our equation.

x = 3 ( 1 ) 2 27 3 = 1
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A quantity x varies directly with the square of y and inversely with z . If x = 40 when y = 4 and z = 2 , find x when y = 10 and z = 25.

x = 20

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Access these online resources for additional instruction and practice with direct and inverse variation.

Visit this website for additional practice questions from Learningpod.

Key equations

Direct variation y = k x n , k  is a nonzero constant .
Inverse variation y = k x n , k  is a nonzero constant .

Key concepts

  • A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See [link] .
  • Two variables that are directly proportional to one another will have a constant ratio.
  • A relationship where one quantity is a constant divided by another quantity is called inverse variation. See [link] .
  • Two variables that are inversely proportional to one another will have a constant multiple. See [link] .
  • In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See [link] .

Section exercises

Verbal

What is true of the appearance of graphs that reflect a direct variation between two variables?

The graph will have the appearance of a power function.

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If two variables vary inversely, what will an equation representing their relationship look like?

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Is there a limit to the number of variables that can vary jointly? Explain.

No. Multiple variables may jointly vary.

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Algebraic

For the following exercises, write an equation describing the relationship of the given variables.

Practice Key Terms 7

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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