Under certain circumstances, a rumor spreads according to the equation $\displaystyle p(t)=\frac{1}{1+ae^{-kt}}$ where $p(t)$ is the proportion of the population that knows the rumor at time $t$, and $a$ and $k$ are positive constants.
(a) Find $\displaystyle\lim_{t \to \infty}p(t)$.
(b) Find the rate of speed of the rumor.
(c) How long will it take for 80% of the population to hear the rumor? Here, take $a=10, \; k=0.5$ with $t$ measured in hours.

(a) limit $=$
(b) $p'(t)=$
(c) hours