Answer
(a) $\lim\limits_{t \to \infty}p(t) = 1$
(b) $\frac{dp}{dt} = \frac{kae^{-kt}}{(1+ae^{-kt})^2}$
(c) It will take about 7.4 hours for 80% of the population to hear the rumor.
Work Step by Step
(a) $p(t) = \frac{1}{1+ae^{-kt}}$
$\lim\limits_{t \to \infty}p(t) = \lim\limits_{t \to \infty}\frac{1}{1+ae^{-kt}}$
$\lim\limits_{t \to \infty}p(t) = \lim\limits_{t \to \infty}\frac{1}{1+a(0)}$
$\lim\limits_{t \to \infty}p(t) = 1$
(b) $\frac{dp}{dt} = \frac{0-(1)(-kae^{-kt})}{(1+ae^{-kt})^2} = \frac{kae^{-kt}}{(1+ae^{-kt})^2}$
(c) $p(t) = \frac{1}{1+10e^{-0.5t}}$
On the graph, we can see that it will take about 7.4 hours for 80% of the population to hear the rumor.
