Calculus I, differentiation

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!

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  1. 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

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