Question

A single individual starts a rumour in a community of 200 people. the rumour spread at a rate proportional to the number of people who have not yet heard the rumor. After 2days 10 people have heard the rumor

1. how many people will have heard the rumor after 5days
2. how long will it take for the rumor to spread to 100 people

Answer 1

If $N$ denotes the number of people who have heard the rumor, then $200-N$ represents the number of people who haven’t heard the rumor.

Then

$$\dfrac{dN}{dt}=k(200-N)$$ $$\dfrac{dN}{200-N}=kdt$$

Integrate

$$\ln|200-N|=-kt+\ln C$$ $$N=200-Ce^{-kt}$$ $$N(0)=200-C=0=>C=200$$ $$N=200-200e^{-kt}$$ $$N(2)=200-200e^{-k(2)}=10$$ $$e^{-k(2)}=0.95$$ $$2k=-\ln0.95$$ $$k=-\dfrac{1}{2}\ln0.95$$ $$N(t)=200-200e^{(\ln0.95)t/2}$$ $$N(t)=200-200(0.95)^{t/2}$$

1.

$$N(5)=200-200(0.95)^{5/2}=24$$

2.

$$N(t)=200-200e^{(\ln0.95)t/2}=100$$ $$e^{(\ln0.95)t/2}=0.5$$ $$(\ln0.95)t=2\ln0.5$$ $$t=\dfrac{2\ln0.5}{\ln0.95}$$ $$t=27\ days$$