Question

The population of a city increases at the rate of $4$% per year. If in time $t$, the population becomes $p$, then equation of $p$ in terms of $t$ is ______

• A. $p=e^{\frac{t}{25}}$
• B. $p=4.e^{\frac{t}{25}}$
• C. $p=c.e^{\frac{t}{25}}$
• D. $p=\frac{1}{25}{e}^{4t}$

Answer 1

Answer: (C)

$p=c.e^{\frac{t}{25}}$

Population $p$ of the city increases at the rate of $4$%
$\therefore \frac{dp}{dt}=\frac{4}{100}p$
$\therefore \frac{dp}{p}=\frac{4}{100}.dt$
Integrate both sides
$\therefore \int \frac{1}{p}dp=\frac{4}{100}.\int dt$
$\therefore \log p=\frac{1}{25}t+\log c$
$\log c$ is constant of integration
$\therefore \log p-\log c=\frac{1}{25}t$
$\therefore \log \left(\frac{p}{c}\right)=\frac{1}{25}t$
$\therefore \frac{p}{c}=.e^{\frac{t}{25}}$
$\therefore p=c.e^{\frac{t}{25}}$