Exercise 5

The logistic equation may be used to model how a rumor spreads through a group of people. Suppose that $p(t)$ is the fraction of people that have heard the rumor on day $t$. The equation $$\begin{gathered} \frac{dp}{dt} = 0.2p(1-p) \end{gathered}$$ describes how $p$ changes. Suppose initially that one-tenth of the people have heard the rumor; that is, $p(0) = 0.1$.

1. What happens to $p(t)$ after a very long time?

2. Determine a formula for the function $p(t)$.

3. At what time is $p$ changing most rapidly?

4. How long does it take before 80% of the people have heard the rumor?