# 5.8 Modeling using variation  (Page 3/14)

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A quantity $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely with the square of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}y=8\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is 4.

$\text{\hspace{0.17em}}\frac{9}{2}\text{\hspace{0.17em}}$

## 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 $\text{\hspace{0.17em}}c,$ cost, varies jointly with the number of students, $\text{\hspace{0.17em}}n,$ and the distance, $\text{\hspace{0.17em}}d.\text{\hspace{0.17em}}$

## Joint variation

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

For instance, if $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ varies directly with both $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z,\text{\hspace{0.17em}}$ we have $\text{\hspace{0.17em}}x=kyz.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ varies directly with $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ and inversely with $z,$ we have $\text{\hspace{0.17em}}x=\frac{ky}{z}.\text{\hspace{0.17em}}$ Notice that we only use one constant in a joint variation equation.

## Solving problems involving joint variation

A quantity $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ varies directly with the square of $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ and inversely with the cube root of $\text{\hspace{0.17em}}z.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}x=6\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}y=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=8,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}y=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=27.\text{\hspace{0.17em}}$

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

$x=\frac{k{y}^{2}}{\sqrt[3]{z}}$

Substitute $\text{\hspace{0.17em}}x=6,\text{\hspace{0.17em}}$ $y=2,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=8\text{\hspace{0.17em}}$ to find the value of the constant $\text{\hspace{0.17em}}k.\text{\hspace{0.17em}}$

$\begin{array}{ccc}\hfill 6& =& \frac{k{2}^{2}}{\sqrt[3]{8}}\hfill \\ \hfill 6& =& \frac{4k}{2}\hfill \\ \hfill 3& =& k\hfill \end{array}$

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

$x=\frac{3{y}^{2}}{\sqrt[3]{z}}$

To find $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}y=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=27,\text{\hspace{0.17em}}$ we will substitute values for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ into our equation.

$\begin{array}{ccc}\hfill x& =& \hfill \frac{3{\left(1\right)}^{2}}{\sqrt[3]{27}}\\ & =& 1\hfill \end{array}$

A quantity $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ varies directly with the square of $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ and inversely with $\text{\hspace{0.17em}}z.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}x=40\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}y=4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=2,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}y=10\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=25.$

$\text{\hspace{0.17em}}x=20\text{\hspace{0.17em}}$

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 Inverse variation

## 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] .

## 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.

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

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

No. Multiple variables may jointly vary.

## Algebraic

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