Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation $$\frac{dP}{dt} = c \ln \left( \frac{K}{P} \right) P$$ where $c$ is a constant and $K$ is the carrying capacity.

(a) Solve this differential equation for $c=0.1$, $K=3000$, and initial population $P_0=200$.
$P(t)=$ .

(b) Compute the limiting value of the size of the population.
$\displaystyle \lim_{t\to \infty} P(t) =$ .

(c) At what value of $P$ does $P$ grow fastest?
$P=$ .