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 7700 units to hundreds of units, since the sales $S$ is given in hundreds of units. $7700$ units $= 77$ hundreds of units. We then set the equation greater than $77$: $55.7 - 37.5 \cos \frac { \pi t } { 6 } > 77$. Solving this inequality will give us the range of $t$ (months) when sales exceed 7700 units. Without solving the inequality explicitly, we know from the nature of the cosine function and the given equation that sales will peak in the summer months and be lowest in the winter months due to the negative cosine component. The correct answer must span the peak sales period, which typically includes the summer months. Among the options, May through July (E) is the most plausible period for sales to exceed 7700 units, aligning with expected seasonal variations in the demand for baseball equipment.

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