Answer
The population after 15 years is $1091$
Work Step by Step
Follow the Logistic Growth Models:
$P(t)=\frac{CP_0}{P_0+(C-P_0)e^{-rt}}$
$r=\frac{1}{t_1} \ln[\frac{P_2(P_1-P_0)}{P_0(P_2-P_1)}]$
$C=\frac{P_0[P_1(P_0+P_2)-2P_0P_2]}{P^2_1-P_0P_2}$
From the given problem we have:
$t_1=5$
$P_0=500$
$P_1=800$
$P_2=800$
Substitute and solve for $r$:
$r=\frac{1}{5} \ln[\frac{1000(800-500)}{500(1000-800)}]$
$r=0.22$
Solve for C:
$C=\frac{800[800(500+1000)-2.500.1000]}{800^2_1-500.1000}$
$C=1142.86$
Thus,
$P(t)=\frac{1142.86\times500}{500+(1142.86-500)e^{-0.22\times15}}$
$P(t)=1091$
The population after 15 years is $1091$
