Models & Qualitative Behavior · Superchallenge problem

A population follows the logistic model $P' = \dfrac12P\left(1 - \dfrac{P}{K}\right)$ ($t$ in years) and passes from near $0$ up to near $K$. Its growth rate $P'$ rises to a maximum and falls again. For how many years is the growth rate at least half of its maximum? Give an exact answer.