This section draws on and extends the concepts from the preceding exercise, Direct and Inverse Variation.

A Typical Proportionality Problem; Finding the Constant of Proportionality


Write an equation for:
‘$A\,$ is proportional to the square of $\,t\,$, and inversely proportional to the cube of $\,x\,$’

If $\,A = 3\,$ when $\,t = 1\,$ and $\,x = 2\,$, find the constant of proportionality.

What is the value of $\,A\,$ when $\, t = -1\,$ and $\,x = 4\,$?


$\displaystyle A = k\cdot \frac{t^2}{x^3}$ Write the equation that describes the relationship between the variables,
using the information from Direct and Inverse Variation.
Don't forget the constant of proportionality!
$\displaystyle 3 = k\cdot\frac{1^2}{2^3}\,$,     $\displaystyle 3 = \frac{k}{8}$ substitute the known values of $\,A\,$, $\,t\,$ and $\,x\,$;
$k = 24$ solve for the constant of proportionality, $\,k\,$
$\displaystyle A = 24\cdot\frac{t^2}{x^3}\,$,     $\displaystyle A= \frac{24t^2}{x^3}$ the final equation can be written in slightly different ways
$\displaystyle A = \frac{24\cdot (-1)^2}{4^3} = \frac{24}{64} = \frac{3}{8}$ now, whenever any two of the three variables are known,
the remaining variable can be determined
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
What is the graph of $\,y = f(x)\,$?
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.