Question

A rumor spreads through a population in such a way that "t" hours after the rumor starts, the percent of people involved in passing it on is given by P(t)=100[e^(-t)-e^(-4t)]. What is the highest percent of people involved in spreading the rumor within the first 3 h? When does this occur?

Answer 1

Differentiate P(t) to obtain the derivative P'(t).

P'(t) = -100 e^-t + 400 e^(-4t)

The maximum will occur where P'(t) = 0

e^-t = 4e^-4t
1 = 4 e^-3t

There will be one such point within 3 hours.