Chapter 4, Problem 41T

The population $P\left(t\right)$ of a herd of deer on an island can be modeled by $P\left(t\right)=\frac{1200}{1+2e^{-0.12t}}$, where t represents the number of years since the park service has been tracking the herd.

a. Evaluate $P\left(0\right)$ and interpret its meaning in the context of this problem.

b. Use the function to predict the deer population after 4 yr. Round to the nearest whole unit.

c. Use the function to predict the deer population after 8 yr.

d. Determine the number of years required for the deer population to reach 900. Round to the nearest year.

e. What value will the term $\frac{2}{e^{0.12t}}$ approach as $t\to \infty$.

f. Determine the limiting value of $P\left(t\right)$.