Chapter 5.3, Problem 91E

To determine

To find: the months in which sales exceed 7500 units.

Answer to Problem 91E

The sales will exceed 7500 units in January, November and December.

Explanation of Solution

Given information:

The monthly sales $S$ of skiing equipment at a sports store are approximated by

  $S=58.3+32.5\cos \frac{\pi t}{6}$

Where $t$ is the time, with $t=1$ corresponding to January.

Calculation:

The monthly sales $S$ is $S=58.3+32.5\cos \frac{\pi t}{6}$ where $t$ is the time in months.

When sales exceed 7500 units, then the monthly sales will be

  $\begin{array}{c}S=58.3+32.5\cos \frac{\pi t}{6}\text{original}\text{equation}\\ \text{75<58}\text{.3+32}\text{.5}\cdot \text{cos}\left(\frac{\pi t}{6}\right)\text{Subtract}\\ \text{16}\text{.7<32}\text{.5}\cdot \text{cos}\left(\frac{\pi t}{6}\right)\text{Divide}\\ \text{0}\text{.514<cos}\left(\frac{\pi t}{6}\right)\text{Simplify}\end{array}$

Use inverse function and have $\frac{\pi t}{6}>\arccos \left(0.514\right)\Rightarrow t>\frac{6}{\pi }\cdot \arccos \left(0.514\right)$ . So, $t=1,11,12$ .

Therefore, the sales will exceed 7500 units in January, November and December.