is proportional to the number of people who have not yet heard about it. Suppose
that the size of a community is 15,000, that to begin with no one has heard about
a product, but that after 6 days 1500 people know about it. How long will it take
for 2700 people to have heard of it?
SOLUTION:
Let N(t) be the number of people aware of the product at time t. Then we are
given that
N′(t) = k [15,000 − N(t)],which is Case IIa. The solution of this d.e. is
N(t) = 15,000 − ce−kt.Since N(0) = 0, c = 15,000 and
N(t) = 15,000(1 − e−kt).Since 1500 people know of the product after 6 days, we have
We now seek t when N = 2700: