Okay I have been doing this for a while now and I got t = 3, but clearly it was wrong because it actually equals 1.6 days.
I set up the model like this
dp/dt = k(p)^1/2
This is the problem
Let t denote time (in days) and let P denote the population of some species at time t. Suppose that the birth rate β (number of births per day per unit of population) is proportional to P^1/2 and that the death rate δ (number of deaths per day per unit of population) equals 0. Suppose the initial population is 4, and after one day the population is 16. When will the population equal 100?
okay I'm guessing i setup the model correctly
So, I took the integrals of (1/(p)^(1/2)) dP = k dt
and I got 2p^(1/2) = kt + c
p = ((kt/2) + c)^2
c = Po^1/2 (initial population)
so i plugged in the initial values to find the constant k value
16 = ( k/2 + (4)^(1/2))^2 ... Took the square root of both sides
4 = (k/2 + 2)
2 = k/2
4 = k
then for the new value of P (100) i plugged in the values
100 = ( 4t/2 + (4)^1/2)^2
10 = 2t +2
8 = 2t
4 = t
I actually got 4, but the Professor said it was equal to 1.6 days.