a variation on the population problem is when the population size has an inhibitory effect on the growth of the population. hence, if x is the population size then $\large \frac{dx}{dt} = k(a-x)x$. this kind of model can be extended to many situations, called self-limiting problems. one person in a small town with a population of 200 proceeds to spread a rumor. if the rate at which the rumor spreads is proportional to the number of people who know the rumor (x) as well as the number of people who dont know, find an expression for the number of people who know about the rumor after t days.

Question

lgbasallote · Answer

fifty people know about the rumor after one day. how many people will have heard of the rumor after 2 days? will the entire town eventually hear the rumor? Ans. 192

a = total population
k  = constant of proportionality

lgbasallote · Answer

okay...i have no clue how to start this...any hints?

unklerhaukus · Answer

first order exponential growth

lgbasallote · Answer

yeah..what else?

lgbasallote · Answer

i took the integral of both and ended up with $$\ln (\frac{x}{a-x}) = akt$$
is that right?

anonymous · Answer

"+C" on the right.... but other than that.... ezguud...

lgbasallote · Answer

okay so what d i do from there?

lgbasallote · Answer

@Mimi_x3 do you get it?