Example 6.4.2 Revenue

Avery owns a local organic jam company that currently sells about \(1500\) jars a month at a price of \(\$13\) per jar. Avery has found that every time they raise the price by \(25\) cents a jar, they will sell \(50\) fewer jars of jam each month.

In general, this company's revenue can be calculated by multiplying the cost per jar by the total number of jars of jam sold.

If we let \(x\) represent the number of \(25\)-cent increases in the price, then the price per jar will be the current price of thirteen dollars/jar plus \(x\) times \(0.25\) dollars/jar, or \(13+0.25x\text{.}\)

Continuing with \(x\) representing the number of \(25\)-cent increases in the price, we know the company will sell \(50\) fewer jars each time the price increases by \(25\) cents. The number of jars the company will sell will be the \(1500\) they currently sell each month, minus \(50\) jars times \(x\text{,}\) the number of price increases. This gives us the expression \(1500-50x\) to represent how many jars the company will sell after \(x\) \(25\)-cent price increases.

Combining this, we can now write a formula for our revenue model:

\begin{align*} \text{revenue} \amp= \left(\text{price per item}\right)\left(\text{number of items sold}\right)\\ R \amp= \left(13+0.25x\right)\left(1500-50x\right) \end{align*}

To simplify the expression \(\left(13+0.25x\right)\left(1500-50x\right)\text{,}\) we'll need to multiply \(13+0.25x\) by \(1500-50x\text{.}\) In this section, we'll learn how to multiply these two expressions that each have multiple terms.