1.15.2 Modeling Equations Using Direct Variation

Modeling Situations Using Direct Variation 

When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss direct variation here.

Lindsay gets paid $15 per hour at her job. If we let s be her salary, in dollars per hour, and h be the number of hours she has worked, we could model this situation with the equation

$s=15h$

Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.

In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates $x$ and $y$.

Here’s how to model an equation using direct variation for application problems.

Let’s look at an application problem:

The cost of a pizza, $y$, varies directly with its diameter, $x$, and an 8-inch pizza costs $12. What is the equation that relates $x$ and $y$?

Step 1 - Write the formula for direct variation.

$y=kx$

Step 2 - Substitute the given values for the variables.

$12=k(8)$

Step 3 - Solve for the constant of variation.

$k=\frac{12}{8}=\frac{3}{2}$

Step 4 - Write the equation that relates x and y using the constant of variation.

$y=\frac{3}{2}x$