Question

The sum of Rs. $1000$ is compounded continuously, the nominal rate of interest being four percent per annum. In how many years will the amount be twice the original principal? $({\log }_{e}2=0.6931)$.

Answer 1

Let Rs. $x$ be the principal.
The required differential equation is
$\frac{dx}{dt}=\frac{4}{100}x$ (or) $\frac{dx}{x}=0.04dt$
Integrating both sides, we get
$\int \frac{dx}{x}=0.04\int dt$

$\Rightarrow \log x$ $=0.04t+\log c\Rightarrow x=c{e}^{0.04t}$ ....... (1)
Given, $x=1000$, when $t=0$
Hence (1) becomes $c=1000$
when $x=2000$

$\Rightarrow 2000$ $=1000e^{-0.04t}$

$\Rightarrow 2=e^{-0.04t}$
$\Rightarrow {\log }_{e}2=0.04t$
$\Rightarrow t=\frac{{\log }_{e}2}{0.04}=\frac{0.6931}{0.04}=17.32=17$ years
Thus the number of years required for the amount to be twice the principal is approximately $17$ years.