# 5.8 Modeling using variation

 Page 1 / 14
In this section, you will:
• Solve direct variation problems.
• Solve inverse variation problems.
• Solve problems involving joint variation.

A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn$736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.

## Solving direct variation problems

In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formula $\text{\hspace{0.17em}}e=0.16s\text{\hspace{0.17em}}$ tells us her earnings, $\text{\hspace{0.17em}}e,\text{\hspace{0.17em}}$ come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See [link] .

$\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ , sales price $e=0.16s$ Interpretation
$4,600 $e=0.16\left(4,600\right)=736$ A sale of a$4,600 vehicle results in $736 earnings.$9,200 $e=0.16\left(9,200\right)=1,472$ A sale of a $9,200 vehicle results in$1472 earnings.
$18,400 $e=0.16\left(18,400\right)=2,944$ A sale of a$18,400 vehicle results in $2944 earnings. Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from$4,600 to $9,200, and we double the earnings from$736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called direct variation . Each variable in this type of relationship varies directly with the other. [link] represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formula $\text{\hspace{0.17em}}y=k{x}^{n}\text{\hspace{0.17em}}$ is used for direct variation. The value $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is a nonzero constant greater than zero and is called the constant of variation . In this case, $\text{\hspace{0.17em}}k=0.16\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}n=1.\text{\hspace{0.17em}}$ We saw functions like this one when we discussed power functions. ## Direct variation If $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ are related by an equation of the form $\text{\hspace{0.17em}}y=k{x}^{n}\text{\hspace{0.17em}}$ then we say that the relationship is direct variation and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly with, or is proportional to, the $\text{\hspace{0.17em}}n\text{th}\text{\hspace{0.17em}}$ power of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ In direct variation relationships, there is a nonzero constant ratio $\text{\hspace{0.17em}}k=\frac{y}{{x}^{n}},\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is called the constant of variation , which help defines the relationship between the variables. Given a description of a direct variation problem, solve for an unknown. 1. Identify the input, $\text{\hspace{0.17em}}x,$ and the output, $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ 2. Determine the constant of variation. You may need to divide $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ by the specified power of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to determine the constant of variation. 3. Use the constant of variation to write an equation for the relationship. 4. Substitute known values into the equation to find the unknown. ## Solving a direct variation problem The quantity $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly with the cube of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}y=25\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is 6. The general formula for direct variation with a cube is $\text{\hspace{0.17em}}y=k{x}^{3}.\text{\hspace{0.17em}}$ The constant can be found by dividing $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ by the cube of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ $\begin{array}{ccc}\hfill k& =& \frac{y}{{x}^{3}}\hfill \\ & =& \frac{25}{{2}^{3}}\hfill \\ & =& \frac{25}{8}\hfill \end{array}$ Now use the constant to write an equation that represents this relationship. $y=\frac{25}{8}{x}^{3}$ Substitute $\text{\hspace{0.17em}}x=6\text{\hspace{0.17em}}$ and solve for $\text{\hspace{0.17em}}y.$ $\begin{array}{ccc}\hfill y& =& \frac{25}{8}{\left(6\right)}^{3}\hfill \\ & =& 675\hfill \end{array}$ #### Questions & Answers sin^4+sin^2=1, prove that tan^2-tan^4+1=0 SAYANTANI Reply what is the formula used for this question? "Jamal wants to save$54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
what is sine?
what is the standard form of 1
1×10^0
Akugry
Evalute exponential functions
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
factoring polynomial