A rumor spreads through a school. Let y(t) be the fraction of the population that has heard the rumor at time t and assume that the rate at which the rumor spreads is proportional to the product of

Question

the fraction y of the population that has heard the rumor and the fraction 1−y that has not yet heard the rumor.

The school has 1000 students in total. At 8 a.m., 107 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.

90% of the students have heard the rumor after about.______???? hours.

Answer 1

With the logistic model, log-odds change is linear.

The slope of the log-odds is (ln(1/2/(1-1/2)) - ln(107/1000/((1000-107)/1000)))/(12-8) = 1/4 ln(893/107)

Then, the number of hours to from 1/2 to 90% is (ln(.9/.1)-0)/(1/4 ln(893/107)) = 4 ln(9)/ln(893/107) =

4.14227228541836