# 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.

the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
if a=3, b =4 and c=5 find the six trigonometric value sin
pls how do I factorize x⁴+x³-7x²-x+6=0
in a function the input value is called
how do I test for values on the number line
if a=4 b=4 then a+b=
a+b+2ab
Kin
commulative principle
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
x^20+x^15+x^10+x^5/x^2+1
evaluate each algebraic expression. 2x+×_2 if ×=5
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
By the definition, is such that 0!=1.why?
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
hatdog
Mark
jaks
Ryan
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching