Suppose the population $P$ of rainbow trout in a fish hatchery is modeled by the differential equation $$\frac{dP}{dt} = P (3 - P),$$ where $P$ is measured in thousands of trout and $t$ is measured in years. Suppose $P(0) = 1$.

(a) How many trout are initially in the hatchery?
trout.

(b) Find a formula for the population at any time.
$P(t) =$ thousands of trout.

(c) What is the size of the trout population after a very long time?
$\displaystyle \lim_{t \to \infty} P(t) =$ thousands of trout.

(d) At what time is the trout population increasing most rapidly?
$t =$ years.