Calculus I, differentiation
posted by Brian on .
The number, N, of people who have heard a rumor spread by mass media by time, t, is given by N(t)=a(1−e−kt). There are 6 million people in the population, who hear the rumor eventually. If 5% of them heard it on the first day, find the percentage of the population who have heard the rumor after 5 days.
I found N'(t)=(ae^(bx))+axe^(bx)
and i figure a=6M*.05=300000, but i cannot find the %population after 5 days. any help would be awesome, Thanks in advance!

N(t) = a(1  e^kt)
After 1 day, t=1, and we have
.05a = (1  e^k)
.05 = 1  e^k
0.95 = e^k
k = ln 0.95 = 0.0513
so,
N(t) = a(1  e^0.0513t)
So, by the end of day 5,
N(5) = a(1  e^.2565) = .226a = 22.6%
have heard the rumor