The monthly sales S (in hundreds of units)of baseball equipment for an Internet sporting goods site are approximated by $S = 55.7 - 37.5 \cos \frac { \pi t } { 6 }$ where t is the time (in months),with t = 1 corresponding to January.Determine the months when sales exceed 7700 units at any time during the month.

Question

Quizplus · Accepted Answer

To find when sales exceed 7700 units, we first convert the sales figure to the scale used in the equation by dividing 7700 by 100, as the sales S is given in hundreds of units. This gives us $S > 77$. We then set up the inequality:$$55.7 - 37.5 \cos \frac{\pi t}{6} > 77$$Solving for $\cos \frac{\pi t}{6}$, we get:$$-37.5 \cos \frac{\pi t}{6} > 21.3$$$\cos \frac{\pi t}{6} < -\frac{21.3}{37.5}$$\cos \frac{\pi t}{6} < -0.568$This inequality is satisfied when $\frac{\pi t}{6}$ is in the second or third quadrant of the unit circle, where cosine is negative. This corresponds to $t$ being in the range of 3 to 9 months into the year, which is March through September. However, since the question asks for when sales exceed 7700 units, we need to find when the sales start and stop exceeding this threshold within this range.Given the cosine function's behavior and the specific range where the inequality is satisfied, we can determine that sales exceed 7700 units starting just after March and ending just before September, which corresponds to the months of May through July inclusively.Therefore, the correct answer is E) May through July.

The Monthly Sales S (In Hundreds of Units)of Baseball Equipment S=55.7−37.5cos⁡πt6S = 55.7 - 37.5 \cos \frac { \pi t } { 6 }S=55.7−37.5cos6πt​

The Monthly Sales S (In Hundreds of Units)of Baseball Equipment $S = 55.7 - 37.5 \cos \frac { \pi t } { 6 }$