Find the value of an investment of 

$ 10 comma 000$10,000 

for 

1111 

years at an annual interest rate of 

4.55 %4.55%     

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

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

where:
- $ A $ is the future value of the investment,
- $ P $ is the principal (initial amount),
- $ r $ is the annual interest rate (in decimal form),
- $ t $ is the time in years,
- $ e $ is the mathematical constant approximately equal to 2.71828.

Given values:
- $ P = 10,000 $,
- $ r = 4.55\% = 0.0455 $,
- $ t = 1111 $.

Let's calculate this step-by-step. 

#### Step 1: Substitute the values
\[
A = 10,000 \times e^{(0.0455 \times 1111)}
\]

#### Step 2: Calculate the exponent
\[
0.0455 \times 1111 = 50.5505
\]

#### Step 3: Plug into the exponential function
\[
A = 10,000 \times e^{50.5505}
\]

#### Step 4: Calculate $ e^{50.5505} $ and multiply by 10,000
I'll proceed with the calculations.The value of the investment after 1111 years at a 4.55% interest rate, compounded continuously, is approximately:

\[
A \approx 8.99 \times 10^{25}
\]

This result represents an extremely large amount due to the significant compounding effect over such a long time. 

Would you like further details on the calculation or have any questions?

---

1. How does continuous compounding differ from other compounding types?
2. What impact does compounding frequency have on investment growth?
3. Why does the investment value become so large over time?
4. What happens if the interest rate is decreased?
5. How can the formula be adapted for different time periods?

**Tip:** Continuous compounding grows investments fastest due to the constant application of interest, which can lead to very large values over long periods.

Find the value of an investment of $10,000 for 1111 years at an annual interest rate of 4.55% compounded continuously.

Discover how to compute the future value of a $10,000 investment with continuous compounding over 1111 years at an annual interest rate of 4.55%. This problem explores key mathematical concepts like exponential growth and demonstrates the continuous compounding formula.

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