To calculate the compound interest for an investment compounded continuously, we use the formula:

$A = Pe^{rt}$

where:

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

Given:

• $P = 45,000$
• $r = 0.08$ (8% as a decimal)
• $t = 15$

Substituting the values into the formula:

$A = 45000 \times e^{0.08 \times 15}$

Calculating $e^{0.08 \times 15}$:

$0.08 \times 15 = 1.2$ $A = 45000 \times e^{1.2} \approx 45000 \times 3.3201 \approx 149,404.50$

The compound interest is calculated as:

$\text{Compound Interest} = A - P = 149,404.50 - 45,000 = 104,404.50$

Thus, the compound interest is approximately $104,404.50.

Would you like more details on the calculation, or do you have any questions? Here are some related questions you might consider:

1. How does continuous compounding compare to annual compounding?
2. What would the compound interest be if the rate were 6% instead?
3. How does changing the investment period to 10 years affect the interest?
4. What is the formula for calculating compound interest for non-continuous compounding?
5. How would you calculate the total amount if you were to invest monthly instead of as a lump sum?

Tip: For continuous compounding, the rate of interest plays a significant role; even small changes in the rate can lead to significant differences in the final amount.