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

$A = P e^{rt}$

Where:

• $A$ is the future value of the investment,
• $P$ is the principal amount (initial investment),
• $r$ is the annual interest rate (expressed as a decimal),
• $t$ is the time the money is invested for, in years,
• $e$ is the base of the natural logarithm (approximately 2.71828).

Given:

• $P = 4500$,
• $r = 0.031$ (since 3.1% = 0.031),
• $t = 10$.

Now we can plug in the values:

$A = 4500 e^{(0.031)(10)}$

$A = 4500 e^{0.31}$

Calculating the value of $e^{0.31}$:

$e^{0.31} \approx 1.363$

Now, calculating $A$:

$A = 4500 \times 1.363 \approx 6133.50$

Thus, the future value of the investment is approximately $6133.50.

Let me know if you'd like further details or clarifications!

Related Questions:

1. How would the future value change if the interest rate were 5% instead of 3.1%?
2. How do you calculate future value with simple interest?
3. What is the formula for future value when interest is compounded annually instead of continuously?
4. How can you use the compound interest formula with different compounding periods (e.g., monthly)?
5. What is the effect of increasing the number of years on the future value of an investment?

Tip:

When dealing with continuous compounding, remember that the key formula $A = P e^{rt}$ involves the natural exponential function, which is crucial for modeling real-world growth such as population, money, and certain investments.