# Find a mathematical model that represents the statement. (Determine the constant of...

## Question:

Find a mathematical model that represents the statement. (Determine the constant of proportionality.)

y is inversely proportional to {eq}x^3 {/eq}. (y = 5 when x = 2.)

## Proportionality :

Consider any two function {eq}\displaystyle f(y) {/eq} and {eq}\displaystyle f(x) {/eq} who relation is defined either directly proportional or inversely proportional.

If relation is directly propotional then it is represented as {eq}\displaystyle f(y ) \propto f(x) {/eq}

If relation is inversely proportional then it is represented as {eq}\displaystyle f(y) \propto \frac{1}{f(x)} {/eq}

This proportionality is removed by multiplying it with a proportionality costant.

Given {eq}\displaystyle y {/eq} is inversely proportional to {eq}\displaystyle x^3 {/eq}

That gives {eq}\displaystyle y \propto \frac{1}{x^3} {/eq}

Removing proportionality we get

{eq}\displaystyle y =\frac{k}{x^3} {/eq}

where {eq}\displaystyle k {/eq} is a proportionality constant.

Given {eq}\displaystyle y=5 {/eq} when {eq}\displaystyle x=2 {/eq}

Substituting we get

{eq}\displaystyle 5=\frac{k}{2^3} \Rightarrow k = 5(8) {/eq}

So mathematical model is reresented as {eq}\displaystyle y=\frac{40}{x^3} {/eq} 