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 the equation of $p$ in time $t$

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

Answer 1

The correct option is C $p=c{e}^{\frac{3t}{100}}$

$\frac{dp}{dt}=\frac{3p}{100}\Rightarrow \frac{dp}{p}=\frac{3}{100}dt\Rightarrow \ln p=\frac{3}{100}t+C$
$\therefore p=e^{\frac{3t}{100}}\cdot e^C$
$\Rightarrow p=c\cdot e^{\frac{3t}{100}}$ where $c=e^C$