Question

The population of a city increases at the rate $3\%$ per year. If at time t the population of city is p, then find equation of p in time t.

• A. $p=c{e}^{\frac{3t}{100}}$
• B. $p=3e^{\frac{3t}{100}}$
• C. $p=e^{\frac{3t}{100}}$
• D. $p=\frac{3}{100}{e}^{3t}$

Answer 1

Answer: (A)

$p=c{e}^{\frac{3t}{100}}$

p is the population of city at any time t.

Rate of increase of population = 3% per year

= 3% of p

$⟹\frac{dp}{dt}=\frac{3}{100}\times p$

$⟹\frac{dp}{p}=\frac{3dt}{100}$

Integrating above eqauaion, we get

$\int \frac{dp}{p}=\int \frac{3dt}{100}$

$ln\left(p\right)=\frac{3t}{100}+C$

$⟹p=c{e}^{\frac{3t}{100}}$

Hence, the answer is option (A).