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:
[tex]P(t)=P_0e^{rt}[/tex]
Where:
[tex]\begin{gathered} P(t)=\text{ amount in time ''t''} \\ P_0=\text{ initial invesment} \\ r=\text{ percentage rate} \\ t=\text{ time} \end{gathered}[/tex]
Now, we will solve for the value of "r". First, we will divide both sides by P0:
[tex]\frac{P(t)}{P_0}=e^{rt}[/tex]
Now, we take the natural logarithm to both sides:
[tex]\ln(\frac{P(t)}{P_0})=rt[/tex]
Now, we divide both sides by the time "t":
[tex]\frac{1}{t}\ln(\frac{P(t)}{P_0})=r[/tex]
Now, we plug in the values:
[tex]\frac{1}{10}\ln(\frac{25000}{10000})=r[/tex]
Now, we solve the operations:
[tex]0.0916=r[/tex]
This is the interest rate in decimal form. To get the percentage we multiply by 100, and we get:
[tex]9.16\%=r[/tex]
Therefore, the interest rate is 9.16%
