A rumor spreads through a school.

Question

A rumor spreads through a school. Let  be the fraction of the population that has heard the rumor at time  and assume that the rate at which the rumor spreads is proportional to the product of the fraction  of the population that has heard the rumor and the fraction  that has not yet heard the rumor.

The school has 1000 students in total. At 8 a.m., 92 students have heard the rumor, and by noon, half the school has heard it. Using the logistic model explained above, determine how much time passes before 90% of the students will have heard the rumor.

Answer 1

Let f(t) be the fraction of students who have heard the rumor at time t.

df/dt = kf(1-f)

Use partial fractions to integrate this d.e.

This d.e. has general solution f/(1-f)=Ce^kt.

The initial conditions imply that C=92/908 and k=(1/4)ln(908/92)

Solving the general solution for the time when f=.9 gives t=7.83886

I have not supplied the details of this solution, but if you have SPECIFIC questions, I will try to answer.