A rumor spreads through a small town. The speed that the rumor spreads is the pr

Question

A rumor spreads through a small town. The speed that the rumor spreads is the product of the fraction of people who have heard the rumor and the fraction of people who have not heard the rumor. Do the following: Let y(t) be the fraction of the people in the town who have heard the rumor at time t.. Write a differential equation involving y which models the above situation. Find the general solution of the equation you found in part a. Find a specific solution to the equation under the assumption that at time zero a tenth of the town has heard the rumor. Using your function y(t), determine what percent of people have heard the rumor at time 1 (you may round your answer to two decimals for this part).

Explanation / Answer

a)dy/dt=y*k

where k is fraction of unheard people

k=A-y

where A is total population

=>dy/dt=Ay-y^2

b)=>dy/y(A-y)=dt

upon integrating we get

1/A(ln(y/(A-y))=t+c

=>y/A-y =Ce^At

=>y=ACe^At/(1+Ce^At)

where C is constant

c)at t=0 y=.1A

=>.1A=AC/(1+C)

=>C/1+C=.1

=>C=1/9=.11

=>y=A e^At/(9+e^At)

d)t=1 y=Ae^A/(9+e^A)

%=1-(e^A/(9+e^A))

=9/(9+e^A) %=77%

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