Sure, let's walk through the solution step-by-step.
1. Identify the Known Variables:
- Initial Investment ([tex]\( P \)[/tex]): [tex]$400
- Final Amount After 10 Years (\( A \)): $[/tex]1,605
- Time for Growth ([tex]\( t \)[/tex]): 10 years
2. Formula for Continuously Compounded Interest:
The formula for the continuously compounded interest is:
[tex]\[
A = P \cdot e^{rt}
\][/tex]
Where:
- [tex]\( A \)[/tex] is the final amount
- [tex]\( P \)[/tex] is the initial investment
- [tex]\( r \)[/tex] is the annual interest rate
- [tex]\( t \)[/tex] is the time in years
- [tex]\( e \)[/tex] is the base of the natural logarithm
3. Calculate the Annual Interest Rate:
To find the annual interest rate [tex]\( r \)[/tex], we need to rearrange the formula:
[tex]\[
r = \frac{1}{t} \ln \left( \frac{A}{P} \right)
\][/tex]
- Using the given values:
[tex]\[
r = \frac{1}{10} \ln \left( \frac{1605}{400} \right)
\][/tex]
- This simplifies to:
[tex]\[
r = \ln \left( \frac{1605}{400} \right) / 10
\][/tex]
- Calculating this gives:
[tex]\[
r \approx 0.1389 \text{ (or 13.89% as a percentage)}
\][/tex]
4. Find the Time to Double the Investment:
To find the time ([tex]\( t_d \)[/tex]) it takes to double the investment, use the continuously compounded interest formula with [tex]\( A = 2P \)[/tex]:
[tex]\[
2P = P \cdot e^{rt_d}
\][/tex]
- Simplifying, we get:
[tex]\[
2 = e^{rt_d}
\][/tex]
- Taking the natural logarithm of both sides:
[tex]\[
\ln(2) = rt_d
\][/tex]
- Solving for [tex]\( t_d \)[/tex]:
[tex]\[
t_d = \frac{\ln(2)}{r}
\][/tex]
- Using the previously calculated interest rate [tex]\( r = 0.1389 \)[/tex]:
[tex]\[
t_d = \frac{\ln(2)}{0.1389}
\][/tex]
- Simplifying this gives:
[tex]\[
t_d \approx 4.99 \text{ years}
\][/tex]
### Final Answer:
- The continuously compounded interest rate is [tex]\( 13.89\% \)[/tex].
- The time to double the investment is approximately [tex]\( 4.99 \)[/tex] years.
