Answer
The investment would triple in value after around 20 years.
Work Step by Step
We have an investment with a $5.75\%$ interest compounded continuously. And we would suppose that this $5.75\%$ interest is an annual one.
That means every year, that investment would increase by $5.75\%$ of the amount available.
So, if we take the inital value of the investment to be $a_i$:
- After first year, it would increase by $5.75\%$ of $a_i$ -> now its value is $105.75\%a_i$ or $1.0575a_i$
- After second year, it would increase by $5.75\%$ of $1.0575a_i$ -> its value now is $1.0575\times1.0575a_i=(1.0575)^2a_i$.
Therefore, if we take $t$ to be the amount of time for the investment to triple the money (which equals to $3a_i$), we have this equation:
$$(1.0575)^ta_i=3a_i$$
$$(1.0575)^t=3$$
Here, we use graphing calculator and find out that $$t=19.65\approx20(years)$$
So the investment would triple in value after around 20 years.
