To solve this problem, we need to use the formula for continuously compounded interest, which is:

$A = P e^{rt}$

where:

• $A$ is the amount of money after time $t$,
• $P$ is the initial investment (principal),
• $r$ is the annual interest rate (in decimal form),
• $t$ is the time in years,
• $e$ is Euler's number ($\approx 2.718$).

Step 1: Time to Double the Investment

The problem states that the time to double the investment is 13.86 years with an interest rate of 5%. To verify this, we set $A = 2P$ (since the investment doubles) and solve for $t$:

$2P = P e^{0.05t}$ Dividing both sides by $P$: $2 = e^{0.05t}$ Taking the natural logarithm of both sides: $\ln(2) = 0.05t$ Solving for $t$: $t = \frac{\ln(2)}{0.05} \approx 13.86 \, \text{years}$

So the time to double is correct.

Step 2: Amount After 10 Years

To calculate the amount after 10 years with an initial investment of $5,000 and a 5% interest rate, we use the formula $A = 5000 \cdot e^{0.05 \cdot 10}$:

$A = 5000 \cdot e^{0.5} \approx 5000 \cdot 1.64872 \approx 8243.60$

Thus, the amount after 10 years is approximately $8,243.60.

Step 3: Amount After 25 Years

Similarly, to find the amount after 25 years, we use the formula $A = 5000 \cdot e^{0.05 \cdot 25}$:

$A = 5000 \cdot e^{1.25} \approx 5000 \cdot 3.49034 \approx 17,451.70$

Thus, the amount after 25 years is approximately $17,451.70.

Final Table

Initial Investment	Annual Rate	Time to Double (years)	Amount After 10 Years	Amount After 25 Years
$5,000	5%	13.86	$8,243.60	$17,451.70

Would you like more details on any part of this, or do you have any questions?

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Here are 5 related questions to consider:

1. How would the results change if the interest rate were increased to 6%?
2. How do you calculate the time to triple the investment with continuous compounding?
3. What happens to the growth if the interest is compounded monthly instead of continuously?
4. How would you compute the interest after 15 years instead of 10?
5. How does the formula change if you want to calculate compound interest for an investment that doesn’t grow continuously?

Tip: When calculating interest for continuous compounding, always remember to convert the percentage rate into a decimal (e.g., 5% becomes 0.05).