Question

The number, NN, of people who have heard a rumor spread by mass media by time tt is modeled by N(t)=a(1−e^(−bt)). There are 100,000 people in the population who hear the rumor eventually. If 30% of them heard it the first day, find bb, assuming tt is measured in days. Give your answer to three decimal places.

Answer 1

The given function is

N(t) = a(1-e^-bt)

Given that 100000 people hear the rumor eventually. This means as time t tends to infinity, N=100000, t=infinity

100000=a(1−e^-b∞)

100000 = a(1-(1/e^b∞)) infinity * n = infinity and any number power of infinity is also infinity, 1/infinity is 0.

100000 = a(1-0)

a = 100000

Now 30% 0f people heard the rumor on the first day, so t =1

30% of 100000 = 100000(1-e^-b(1))   

0.30 * 100000 = 100000(1-e^-b)

0.30 = (1-e^-b)

e^-b = 1-0.30

e^-b = 0.70

ln(e^-b) = ln(0.70)

-b = -0.356

b = 0.356.