Question

Annabell invested $2000 into an account earning 3.5% interest compounded continuously. How long will it take to double her investment?

Answer 1

$\ 2000$ will become $\ 4000$ in 19.80 years that is approximately 20 years when compounded continuously at the annual interest rate of $3.5\%$

Solution:

Given that  

Amount investe by Annabell = $\ 2000$,

Rate if interest $= 3.5\% = 0.035$  

$Required \ amount = double \ of \ investment = 2\times \ 2000 = \ 4000$

And most important thing that interest is compounded continuously . Formula of Amount where interest is compounded continuously is as follows ,

$\mathrm{A}=\mathrm{P} e^{\mathrm{r}{t}}$

Where A is final amount,  

P is principal Amount,

r = rate of interest  

And t = duration in years

In our case $A = \ 4000; \ P = \ 2000; \ r = 0.035$  

Need to evaluate t that is number of year.

On substituting given values in formula of amount we get

$\begin{array}{l}{4000=2000 e^{0.035 t}} \\\\ {=>\frac{4000}{2000}=e^{0.035 t}} \\\\ {\Rightarrow 2=e^{0.035 t}}\end{array}$

Taking log both the sides,  

$\ln (2)=0.035 \mathrm{t} \times \ln (\mathrm{e})$

$\Rightarrow \mathrm{t}=\frac{\ln (2)}{0.035}==19.80$

That is approximately 20 years.