Question

The rate of growth of population of a city at any time is proportional to the size of the population at that time. For a certain city, the consumer of proportionality is $0.04$. The population of the city after $25$ years, if the initial population is $10,000$ is $(e=2.7182)$.

• A. $27182$
• B. $27164$
• C. $27000$
• D. $27272$

Answer 1

The correct option is A $27182$

Let $P\left(t\right)$ be the population of the city after $t$ years.

Given, $\frac{dP\left(t\right)}{dt}=0.04P\left(t\right)\Rightarrow \frac{dP\left(t\right)}{P\left(t\right)}=0.04dt$

Integrating on both sides,

$\ln P\left(t\right)=0.04t+\ln C$ where $\ln C$ is the constant of Integration.

On taking antilog,

$P\left(t\right)=C{e}^{0.04t}$

When $t=0\Rightarrow P\left(0\right)=10000=C$

So, $P\left(25\right)=10000e^{0.04\times 25}=10000e\approx 27182$

ie, Option A is correct answer.