# PROPORTIONALITY PROBLEMS

• PRACTICE (online exercises and printable worksheets)

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

QUESTION:

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 $\,x = 4\,$?

SOLUTION:

 $\displaystyle A = k\cdot \frac{t^2}{x^3}$ 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 $\,x\,$; simplify $k = 24$ 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
What is the graph of $\,y = f(x)\,$?