Chapter 4.3, Problem 60E

Solution

To identify: The number of Get Wet swimsuits sold in January, April, June, October, and December. Get Wet, Inc., manufactures swimwear, a seasonal product. The monthly sales x (in thousands) for Get Wet swimsuits are modeled by the equation $x=72.4+61.7\sin \frac{\pi t}{6}$ .

For which two of these months are projected sales the same?

The sales were same for months of June and December because of periodic behavior of sine function

Given information

The given sale of swimwear is modelled by following equation

  $x=72.4+61.7\sin \frac{\pi t}{6}$

Here t represents the number of month of the year.

That is t=1 corresponds the month January and $t=2$ corresponds the month February and so on.

Explanation:

Consider the given information

Sale in January

Put the value of $t=1$ in the equation.

  $\begin{array}{c}x=72.4+61.7\sin \frac{\pi t}{6}\\ x=72.4+61.7\sin \frac{\pi \cdot 1}{6}\\ x=72.4+61.7\cdot \frac{1}{2}\\ x=72.4+30.85\\ =103.25\end{array}$

Therefore $\text{103}\text{.25}\text{thousands}$ of swimsuit are sold in the month of January.

Sale in April

Put the value of $t=4$ in the equation.

  $\begin{array}{c}x=72.4+61.7\sin \frac{\pi t}{6}\\ x=72.4+61.7\sin \frac{\pi \cdot 4}{6}\\ x=72.4+61.7\cdot \frac{\sqrt{3}}{2}\\ x=72.4+53.43\\ =125.83\end{array}$

Therefore $\text{125}\text{.83}\text{thousands}$ of swimsuit are sold in the month of April.

Sale in June

Put the value of $t=6$ in the equation.

  $\begin{array}{c}x=72.4+61.7\sin \frac{\pi t}{6}\\ x=72.4+61.7\sin \frac{\pi \cdot 6}{6}\\ x=72.4+61.7\cdot 0\\ x=72.4\end{array}$

Therefore $\text{72}\text{.4}\text{thousands}$ of swimsuit are sold in the month of June.

Sale in October

Put the value of $t=10$ in the equation.

  $\begin{array}{c}x=72.4+61.7\sin \frac{\pi t}{6}\\ x=72.4+61.7\sin \frac{\pi \cdot 10}{6}\\ x=72.4+61.7\cdot \left(-\frac{\sqrt{3}}{2}\right)\\ x=72.4-55.43\\ =20.97\end{array}$

Therefore $\text{20}\text{.97}\text{thousands}$ of swimsuit are sold in the month of October.

Sale in December

Put the value of $t=12$ in the equation.

  $\begin{array}{c}x=72.4+61.7\sin \frac{\pi t}{6}\\ x=72.4+61.7\sin \frac{\pi \cdot 12}{6}\\ x=72.4+61.7\cdot 0\\ x=72.4\end{array}$

Therefore $\text{72}\text{.4}\text{thousands}$ of swimsuit are sold in the month of December.

Hence sales were same for months of June and December because of periodic behavior of sine function.