Question

If the rate of increase of the population of the city is $5\%$ per year. In t time population of city is P, then expression of P in terms of t is _________.

• A. $P=3\cdot e^{t/20}$
• B. $P=e^{t/20}$
• C. $P=c\cdot e^{t/20}$
• D. $P=\frac{1}{20}{e}^{5t}$

Answer 1

The correct option is C $P=c\cdot e^{t/20}$

Suppose population is $P_0$
$\therefore$ During time t it will be P(t)
Also the rate of increase of the population is $5\%$
$\therefore \frac{dp}{dt}=\frac{5}{10v}P$
$\therefore \frac{dp}{p}=\frac{1}{20}dt$
$\therefore \int \frac{1}{p}dp=\frac{1}{20}\int dt$
$\therefore logP=\frac{t}{20}+logc$
$\therefore logP-logc=\frac{t}{20}$
$\therefore log\left(\frac{P}{c}\right)=\frac{t}{20}$
$\therefore \frac{P}{c}=c{e}^{\frac{t}{20}}$
$\therefore P=c_{\infty }{e}^{\frac{t}{20}}$.