Chapter 7.5, Problem 5E

To determine

To calculate : the population after another three years

Answer to Problem 5E

The population after another three years is $9000$

Explanation of Solution

Given information : The question is

A population grows according to a logistic model with initial population $1000$ and carrying capacity $10,000$ and the population after one year is $2500$

Calculation : The logistic equation is

  $\begin{array}{l}\frac{dP}{dt}=kP\left(1-\frac{P}{1000}\right)\\ P=\frac{M}{1+A{e}^{-k}}\\ \text{Where, }A=\frac{M-P_0}{P_0}\\ A=\frac{10000-1000}{1000}\\ A=9\\ P=\frac{10000}{1+9e^{-k}}\\ 2500=\frac{10000}{1+9e^{-k}}\\ 2500+2500\left(9e^{-k}\right)=10000\\ 2500\left(9e^{-k}\right)=7500\\ e^{-k}=\frac{3}{9}\\ \ln e^{-k}=\ln \frac{1}{3}\\ k=-\ln \frac{1}{3}\\ k=1.0986\\ P=\frac{10000}{1+9e^{-1.0986}}\\ t=4\\ P\left(4\right)=\frac{10000}{1+9e^{-1.0986\left(4\right)}}\\ P\left(4\right)=8999.96\\ P\left(4\right)\approx 9000\end{array}$