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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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A quantity 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 vary jointly? 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

(x2-2x+8)-4(x2-3x+5)
Ayush Reply
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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
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
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
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Miranda
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Jhovie
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Propessor Reply
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Miranda
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sita Reply
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Steve
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answer and questions in exercise 11.2 sums
Yp Reply
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Alaba
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Chris
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Chris
cos (-z)= cos z .
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(x+x)3=?
Narad
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Obed
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liyemaikhaya Reply
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Practice Key Terms 7

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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