Find the value of an investment of

$ 10 comma 000$10,000

for

1414

years at an annual interest rate of

3.15 %3.15%

compounded continuously.

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Part 1

The value of the investment is****

​$enter your response here.

​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

To find the value of an investment compounded continuously, we use the formula for continuous compounding:

\[
A = P \cdot e^{(r \cdot t)}
\]

where:
- $ A $ is the amount of the investment after $ t $ years,
- $ P $ is the principal or initial investment amount,
- $ r $ is the annual interest rate (as a decimal),
- $ t $ is the time in years, and
- $ e $ is the base of the natural logarithm, approximately equal to 2.71828.

### Given Values:
- $ P = 10,000 $
- $ r = 3.15\% = 0.0315 $
- $ t = 14 $ years

### Calculation:
Substitute these values into the formula:

\[
A = 10,000 \cdot e^{(0.0315 \cdot 14)}
\]

1. Calculate the exponent:  
   \[
   0.0315 \times 14 = 0.441
   \]

2. Calculate $ e^{0.441} $, then multiply by 10,000.

Let's perform this calculation.The value of the investment after 14 years, compounded continuously at an annual interest rate of 3.15%, is approximately:

\[
A \approx \$15,542.61
\]

Let me know if you’d like a more detailed breakdown or have any questions.

---

### Related Questions
1. How does continuous compounding compare to annual compounding for the same rate?
2. What would the investment value be if compounded annually instead?
3. How much additional value does continuous compounding add over simple interest?
4. How long would it take for the investment to double with this interest rate?
5. What would be the future value if the interest rate increased to 4%?

**Tip:** Continuous compounding is useful for maximizing returns on investments since it grows the principal at every moment.

Find the value of an investment of $10,000 for 14 years at an annual interest rate of 3.15% compounded continuously.

Discover how to calculate the future value of a $10,000 investment over 14 years with a continuous compounding interest rate of 3.15%. This solution covers the continuous compounding formula and provides a clear, step-by-step calculation. Ideal for high school students and anyone interested in finance or exponential growth.

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