# 5.8 Modeling using variation  (Page 3/14)

 Page 3 / 14

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.

(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
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jai
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Miranda
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jai
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Miranda
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I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
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Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
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jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
Jeffrey
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Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
which equation
Miranda
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Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
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Ye
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Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
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Chinni
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Ranjay
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ANSHU
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Chinni
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Chinni
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Hassan
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SORIE
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Abdel
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Yaona
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SORIE
it's 12