To solve the problem, we will use the formula for continuous compounding:

$A = P e^{rt}$

Where:

• $A$ = final amount
• $P$ = initial principal ($2000)
• $r$ = annual interest rate (8% or 0.08)
• $t$ = time in years (12 years)
• $e$ = Euler's number (approximately 2.71828)

Plugging in the values:

$A = 2000 \times e^{0.08 \times 12}$

$A = 2000 \times e^{0.96}$

Calculation:

1. Compute $e^{0.96}$.
2. Multiply the result by 2000.

Let's perform the calculation:

$A \approx 2000 \times 2.6117 \approx 5223.40$

So, the final value, rounded to the nearest cent, is $5223.40.

Would you like more details or have any questions?

Related Questions:

1. What would be the final value if the interest rate was 10% instead of 8%?
2. How does continuous compounding compare to annual compounding for the same scenario?
3. What would be the total interest earned over the 12 years?
4. How long would it take for the investment to double with continuous compounding at 8%?
5. What formula would you use if the compounding was quarterly instead of continuous?

Tip:

For continuous compounding, the formula $A = Pe^{rt}$ is very efficient, as it doesn't require dividing the rate or adjusting the time for compounding intervals.