The number of people that hear a rumor follows logistic growth. In a school of 1500 students, 5 students start a rumor. After 2 hours, 120 students have heard about the rumor.

Question

Recall: dy/dx=ky(1-Y/L) and y=L/(1+be^(-kt))

I found the logistic growth equation to be 1500/(1+299e^-1.628548519t)

What is the rate of growth when the rumor is spreading the fastest?

Answer 1

y = 1/(1+be^kt)
y' = kbe^kt/(1+be^kt)^2
y' will be greatest when y" = 0
y" = k^2be^kt(3e^kt-1)/(1+be^kt)^3
so, that will be when t = -1/k ln3
In this case, when t = 1.48