Answer to Question #167497 in Differential Equations for Kabilen

Question

The population of a town grows at a rate proportional to the population present at time t. The initial population of 5000 increases by 50% in 10 years. What will be the population in 30 years?

Answer 1

According to the question,

$\frac{dP}{dt}=kP$, where $P$ is population at any time

solving equation,

$\int\frac{dP}{P}=\int kdt$ ,

$lnP=kt+C,$

Applying the conditions,

at $t=0, \space P=5000$

$ln(5000)=C$

at $t=10,\space P=7500$

$ln(7500)=k(10)+ln(500)$

$k=\frac{ln(1.5)}{10}=0,040547$

So Equation of the Population growth is given by

$P=5000e^{0.040547t}$

Population in 30 years will be

$P=5000e^{0.040547 \cdot30}=16875.25\approx 16875$ persons