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.

Question

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!

Steve · Accepted Answer

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