To solve this problem, we'll use the formula for continuously compounded interest:

$A = P e^{rt}$

Where:

• $A$ is the final amount of the investment.
• $P$ is the initial principal (in this case, $1,000).
• $r$ is the annual interest rate (4.5%, or 0.045 in decimal form).
• $t$ is the time in years.
• $e$ is Euler's number, approximately 2.71828.

We will calculate the value of the investment after each time period provided.

(a) After 2 years:

$A = 1000 \cdot e^{0.045 \cdot 2}$

(b) After 4 years:

$A = 1000 \cdot e^{0.045 \cdot 4}$

(c) After 12 years:

$A = 1000 \cdot e^{0.045 \cdot 12}$

Let's calculate each value.The value of the investment after the given number of years is:

(a) After 2 years: $1,094.17
(b) After 4 years: $1,197.22
(c) After 12 years: $1,716.01

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What happens to the investment if the interest rate increases to 6%?
2. How does the formula for compound interest differ when compounded monthly versus continuously?
3. Can you derive the formula for continuously compounded interest from first principles?
4. What would the final amount be after 8 years at the same rate?
5. How does continuously compounding interest compare to annual compounding over time?

Tip: The faster the interest is compounded (e.g., continuously vs. monthly), the higher the final amount will be.