Answer
The population after a total of 4 years is 9000
Work Step by Step
A logistic equation has this form:
$\frac{dP}{dt} = kP(1-\frac{P}{M})$
The solution has this form:
$P(t) = \frac{M}{1+Ae^{-kt}}~~$ where $~~A = \frac{M-P_0}{P_0}$
We can find $A$:
$A = \frac{M-P_0}{P_0} = \frac{10,000-1000}{1000} = 9$
We can write the solution:
$P(t) = \frac{10,000}{1+9e^{-kt}}$
The population grows to 2500 after 1 year. We can find $k$:
$P(t) = \frac{10,000}{1+9e^{-(k)(1)}} = 2500$
$1+9e^{-k} = \frac{10,000}{2500}$
$1+9e^{-k} = 4$
$9e^{-k} = 3$
$e^{-k} = \frac{1}{3}$
$-k = ln(\frac{1}{3})$
$k = -ln(\frac{1}{3})$
$k = 1.1$
We can find the population after a total of 4 years:
$P(4) = \frac{10,000}{1+9e^{-(1.1)(4)}} = 9000$
The population after a total of 4 years is 9000
