# The Logistic Function (Population Growth)

**Calculus**Level 3

\[ P(t)=\frac{A}{1+Be^{-Ct}}, \] where constants \(A\), \(B\), and \(C\) are usually determined experimentally.

Suppose that population in a town is modeled by

\[ P(t)=\frac{20,000}{1+4e^{-2t}}, \] where \(P(t)\) is number of population and \(t\) is time in year. Let \(t_1\) be the time when the population growth rate begin to decline and \(t_2\) be the time when the population reach \(80\) percent of its limit, then \(t_1+t_2\) can be expressed as \(T\ln 2\). Determine the value of \(T\).