Q44. - Algebra 2

Question

ZOOLOGY For Exercises 44–46, use the following information.

The population of predators and prey in a closed ecological system tends to vary periodically over time. In a certain system, the population of owls O can be represented by $O=150+30\sin \left(\frac{\pi }{10}t\right)$ where t is the time in years since January 1, 2001. In that same system, the population of mice M can be represented by $M=600+300\sin \left(\frac{\pi }{10}t+\frac{\pi }{20}\right)$.

Find the maximum number of owls. After how many years does this occur?

Step-by-Step Solution

Answer

The maximum number of owls is 180 and that occurs after 5 years. 

Step 1. Given

The population of owls O can be represented by $O=150+30\sin \left(\frac{\pi }{10}t\right)$.

Step 2. To determine

Find the maximum number of owls. After how many years does this occur?

Step 3. Calculation

The maximum value of sine is 1. 

So, the maximum value of O is: $150+30\left(1\right)=180$.

Sine has maximum value 1 at $\frac{\pi }{2}$.

That means,

 $\begin{array}{l}\frac{\pi }{10}t=\frac{\pi }{2}\\ t=\frac{\pi }{2}\times \frac{10}{\pi }\\ t=5\end{array}$

Hence, the maximum number of owls is 180 and that occurs after 5 years.