Question

Population of a town increase at rate proportional to existing population at that time. If population of town was $2,00,000$ in year 1990 and $2,50,000$ in year 2000.What will be population of town in year 2010?

Answer 1

$\to$ Population, $\frac{dP}{dt}\propto P$

$\Rightarrow \frac{dP}{dt}=kP\Rightarrow \int \frac{dP}{P}=\int Rdt$

$\Rightarrow lnP=kt+c$

let $t=0$ at $1990$

$\Rightarrow ln\left(200000\right)=C$

In $2000,ln\left(250000\right)=10k+ln\left(200000\right)$

$\Rightarrow k=ln\left(\frac{5}{4}\right)\times \frac{1}{10}$

In $2010$,

$lnP=\frac{1}{10}ln\left(\frac{5}{4}\right)\times 20+ln\left(200000\right)$

$=ln(\frac{5^2}{4^2}\times 200000)$

$=ln\left(312500\right)$

$\Rightarrow P=312500$.