Solve Apps: Trigonometric Equations
-The weekly sales in thousands of items of a product has a seasonal sales record approximated by $\mathrm { n } = 84.37 + 15 \sin \frac { \pi \mathrm { t } } { 24 }$ ( $\mathrm { t }$ is time in weeks with $\mathrm { t } = 1$ referring to the first week in the year). During which week(s) will the sales equal 91,870 items?
A) week 4 , week 20 , and week 52
B) week 30 and week 47
C) week 4 and week 47
D) week 21 and week 30

Question

Quizplus · Accepted Answer

The sales will equal 91,870 items when:$$84.37 + 15 \sin \frac { \pi \mathrm { t } } { 24 } = 91.87$$$$15 \sin \frac { \pi \mathrm { t } } { 24 } = 7.5$$$$\frac { \pi \mathrm { t } } { 24 } = \sin^{-1} \frac { 7.5 } { 15 }$$$$\frac { \pi \mathrm { t } } { 24 } = 0.5$$$$\mathrm { t } = \frac { 24 } { \pi } ( 0.5 ) = 3.82$$Since $t$ represents the week number, the sales will equal 91,870 items during week 4 (rounded up from 3.82).To find the other weeks, we can add or subtract 24 from $t$.$$3.82 + 24 = 27.82$$$$27.82 - 24 = 3.82$$Therefore, the sales will also equal 91,870 items during week 20 (rounded up from 27.82) and week 52 (rounded down from 3.82).

Solve Apps: Trigonometric Equations -The Weekly Sales in Thousands of Items

Solve Apps: Trigonometric Equations
-The Weekly Sales in Thousands of Items