Q14. - Algebra 2

Question

In a certain wildlife refuge, the population of field mice can be modelled by $y=3000+1250\sin \left(\frac{\pi t}{6}\right)$, where (y) represents the number of mice and (t) represents the number of months past March 1 of a given year then what is the maximum number of mice and when does this occur?

Step-by-Step Solution

Answer

The maximum number of mice are $4250$ and it will occur when $t=3$ months have passed March 1 of the given year.

Step 1. Write down the given information.

The population of field mice is modelled by the equation $y=3000+1250\sin \left(\frac{\pi t}{6}\right)$, where (y) represents the number of mice and (t) represents the number of months past March 1 of the given year.

Step 2. Calculation.

From the given equation $y=3000+1250\sin \left(\frac{\pi t}{6}\right)$, the maximum number of mice can be evaluated by equating $\sin \left(\frac{\pi t}{6}\right)=1$ because the maximum value of sine function is 1. Therefore,

  $\begin{array}{c}y=3000+1250{\left[\sin \left(\frac{\pi t}{6}\right)\right]}_{\text{Max}}\\ =3000+1250\left(1\right)\\ =4250\end{array}$

Hence, the maximum number of mice are $4250$ and it will occur when $\sin \left(\frac{\pi t}{6}\right)=1$. Therefore,

 $\begin{array}{c}\sin \left(\frac{\pi t}{6}\right)=1\text{ gives,}\\ \frac{\pi t}{6}={\sin }^{-1}\left(1\right)\\ \frac{\pi t}{6}=\frac{\pi }{2}\\ t=3\end{array}$

Hence, the maximum number of mice are $4250$ and it will occur when $t=3$ months have passed.

Step 3. Conclusion.

The maximum number of mice are $4250$ and it will occur when $t=3$ months have passed March 1 of the given year.