# 3.9 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 $\text{\hspace{0.17em}}k.\text{\hspace{0.17em}}$

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

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.

$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ 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 jointly vary? Explain.

No. Multiple variables may jointly vary.

## Algebraic

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

#### Questions & Answers

If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William