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.

For continuous compounding, the formula to calculate the future value $ A $ of an investment is:

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

where:
- $ P $ is the principal (initial 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 2.71828).

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

### Step-by-Step Solution:

1. Substitute the values into the formula:

   \[
   A = 10,000 \cdot e^{(0.0455 \cdot 1111)}
   \]

2. Calculate $ 0.0455 \cdot 1111 $:

   \[
   0.0455 \cdot 1111 = 50.5505
   \]

3. Now, calculate $ e^{50.5505} $.

4. Finally, multiply the result by 10,000 to find the investment value.

Let’s calculate this in detail:The value of the investment after 1111 years at an annual interest rate of 4.55%, compounded continuously, is approximately:

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

This incredibly large value arises from the long time horizon and continuous compounding.

Would you like further details, or do you have any questions?

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Here are some related questions to explore:
1. How would the value change with a different compounding period, such as annual or monthly?
2. What is the impact of doubling the interest rate on this investment?
3. How does compounding affect the investment compared to simple interest over the same time?
4. What if the investment period were shorter, such as 10 or 50 years?
5. How does the investment value grow over time with different initial amounts?

**Tip:** Continuous compounding grows investments faster due to the constant reinvestment of earnings, especially notable over extended periods.

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

Calculate the value of a $10,000 investment after 1111 years with an annual interest rate of 4.55%, compounded continuously. This solution covers continuous compounding, exponential growth, and relevant financial formulas. Suitable for advanced high school students.

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