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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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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 jointly vary? 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.

Questions & Answers

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
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
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
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
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
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
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Practice Key Terms 7

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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