Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-The weekly sales in thousands of items of a product has a seasonal sales record approximated by $\mathrm { n } = 84.77 + 17.9 \sin \frac { \pi \mathrm { t } } { 24 }$ ( $\mathrm { t }$ is time in weeks with $\mathrm { t } = 1$ referring to the first week in the year). During which week(s) will the sales equal 93,720 items?

Question

Quizplus · Accepted Answer

The equation to solve is $84.77 + 17.9 \sin \frac { \pi \mathrm { t } } { 24 } = 93.72$. Subtracting 84.77 from both sides gives $17.9 \sin \frac { \pi \mathrm { t } } { 24 } = 8.95$. Dividing both sides by 17.9 gives $\sin \frac { \pi \mathrm { t } } { 24 } = 0.5$. The solutions to this equation over the interval [0, $ 2 \pi )$ are $\frac { \pi } { 6 }$ and $\frac { 5 \pi } { 6 }$. Converting these to weeks gives week 4 and week 20. The equation also has a solution at $\frac { 3 \pi } { 2 }$, which corresponds to week 52.

Solve the Equation for Solutions Over the Interval [0 [0,2π)[ 0,2 \pi )[0,2π)

Solve the Equation for Solutions Over the Interval [0 $[ 0,2 \pi )$