Mathematics

This is a logistic growth problem where you need to find the maximum growth rate, not the maximum population. The key insight is that $\frac{dP}{dt}$ itself is a function of $P$—treat it as a single-variable function and find where its derivative equals zero. Since $\frac{dP}{dt} = 0.08P(500-P)$ is a parabola in $P$, its peak occurs at the midpoint between its roots.

This is a logistic growth problem where you need to find the maximum growth rate, not the maximum population. The key insight is that $\frac{dP}{dt}$ itself is a function of $P$—treat it as a single-variable function and find where its derivative equals zero. Since $\frac{dP}{dt} = 0.08P(500-P)$ is a parabola in $P$, its peak occurs at the midpoint between its roots.