Solve the problem.
-The weekly sales in thousands of items of a product has a seasonal sales record approximated by $\mathrm { n } = 84.05 + 15.8 \sin \frac { \pi \mathrm { t } } { 24 }$ ( $\mathrm { t }$ is time in weeks with $\mathrm { t } = 1$ referring to the first week in the year). During which week(s) will the sales equal 91,950 items?
A) week 4, week 20, and week 52
B) week 21 and week 30
C) week 4 and week 47
D) week 30 and week 47

Question

Quizplus · Accepted Answer

The sales will equal 91,950 items when:$$84.05 + 15.8 \sin \frac { \pi \mathrm { t } } { 24 } = 91.95$$$$\sin \frac { \pi \mathrm { t } } { 24 } = \frac { 91.95 - 84.05 } { 15.8 } = 0.5$$$$\frac { \pi \mathrm { t } } { 24 } = \sin^{-1} ( 0.5 ) = \frac { \pi } { 6 }$$$$\mathrm { t } = \frac { 24 } { \pi } \cdot \frac { \pi } { 6 } = 4$$So, the sales will equal 91,950 items during week 4.We can also find the other weeks when the sales will equal 91,950 items by adding or subtracting 24 weeks from week 4.$$4 + 24 = 28$$$$4 - 24 = -20$$Since we are only interested in weeks within the year, we can ignore the negative value.So, the sales will also equal 91,950 items during week 28 and week 52.

Solve the Problem n=84.05+15.8sin⁡πt24\mathrm { n } = 84.05 + 15.8 \sin \frac { \pi \mathrm { t } } { 24 }n=84.05+15.8sin24πt​

Solve the Problem $\mathrm { n } = 84.05 + 15.8 \sin \frac { \pi \mathrm { t } } { 24 }$