To find the future value of an investment with continuous compounding, we use the formula:

$A = P e^{rt}$

where:

• $A$ is the amount of money accumulated after $t$ years, including interest.
• $P$ is the principal amount (the initial amount of money).
• $r$ is the annual interest rate (decimal).
• $t$ is the time the money is invested for in years.
• $e$ is the base of the natural logarithm (approximately equal to 2.71828).

Given:

• $P = 8000$ dollars
• $r = 0.065$ (6.50%)
• $t = 2$ years for part 1 and $t = 20$ years for part 2

Part 1: After 2 years

Substitute the values into the formula:

$A = 8000 \times e^{0.065 \times 2}$

First, calculate the exponent:

$0.065 \times 2 = 0.13$

Now, calculate $e^{0.13}$:

$e^{0.13} \approx 1.13883$

Next, multiply this by the principal amount:

$A = 8000 \times 1.13883 \approx 9110.64$

So, after 2 years, the investment will be worth approximately $\$9110.64$.

Part 2: After 20 years

Substitute the values into the formula:

$A = 8000 \times e^{0.065 \times 20}$

First, calculate the exponent:

$0.065 \times 20 = 1.3$

Now, calculate $e^{1.3}$:

$e^{1.3} \approx 3.6693$

Next, multiply this by the principal amount:

$A = 8000 \times 3.6693 \approx 29354.40$

So, after 20 years, the investment will be worth approximately $\$29354.40$.

Summary

• After 2 years, the investment will be worth approximately $\$9110.64$.
• After 20 years, the investment will be worth approximately $\$29354.40$.

If you want details or have any questions, please let me know.

Questions to expand your understanding:

1. What is the formula for continuous compounding interest?
2. How does the value of $e$ affect continuous compounding?
3. What is the effect of increasing the interest rate on the future value?
4. How does the time period impact the future value in continuous compounding?
5. What are the differences between continuous compounding and annual compounding?
6. How do you calculate the value of $e$ raised to a given power?
7. What would be the future value for different principal amounts?
8. How can you calculate the time needed for an investment to double with continuous compounding?

Tip:

Remember that in continuous compounding, the interest is calculated and added to the principal at every possible instant, leading to slightly higher returns compared to standard compounding methods.