Say I have (1/(Ps - Pu))ln[(P - Pu)/(Ps - P)] = (r/k)t + C.

I'm trying to find t.

Provided the values; The constant eort model also has r = 0:3 per year, K = 3000 kilotons and PS = 1600 kilotons but PU = 0 kilotons. Assuming the same initial condition of P = 1410 when t = 0 calculate the time taken for the population to reach P = 1500 for this model.

[1/(1600 - 0)]ln[(1410 - 0)/(1600 - 1410)] = (0.3/3000)0 + C

(1/1600)*ln(1410/190) = C ---> C = ln(1410/190) /1600

So to find t so that P(t) = 1500 we have

(1/1600)ln(1500/100) = (0.3/3000)t + [(1/1600)*ln(1410/190)

0.00169253137 = (0.3/3000)*t + 0.00125270057

-0.00125270057 -0.00125270057

0.0004398308 = (0.3/3000)t

t = (3000/0.3) * 0.0004398308 = 4.40 years.