Question

An investment of $10,000 is compounded continuously. What annual percentage rate will produce a balance of $25,000 in ten years?

Answer 1

We are given that $10 000 is continuously compounded, and we are asked to determine the rate of change that will produce $25 000 in ten years. To do that we will use the following formula:

$P(t)=P_0e^{rt}$

Where:

$\begin{gathered} P(t)=\text{ amount in time ''t''} \\ P_0=\text{ initial invesment} \\ r=\text{ percentage rate} \\ t=\text{ time} \end{gathered}$

Now, we will solve for the value of "r". First, we will divide both sides by P0:

$\frac{P(t)}{P_0}=e^{rt}$

Now, we take the natural logarithm to both sides:

$\ln(\frac{P(t)}{P_0})=rt$

Now, we divide both sides by the time "t":

$\frac{1}{t}\ln(\frac{P(t)}{P_0})=r$

Now, we plug in the values:

$\frac{1}{10}\ln(\frac{25000}{10000})=r$

Now, we solve the operations:

$0.0916=r$

This is the interest rate in decimal form. To get the percentage we multiply by 100, and we get:

$9.16\%=r$

Therefore, the interest rate is 9.16%