A person places $62700 in an investment account earning an annual rate of 2.6%, compounded continuously. Using the formula V, equals, P, e, start superscript, r, t, end superscriptV=Pe 
rt
 , where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 18 years.

To calculate the amount of money in the account after 18 years using the formula for continuous compounding:

\[
V = P \cdot e^{rt}
\]

where:
- $ P = 62700 $ is the principal amount,
- $ r = 0.026 $ is the annual interest rate (2.6% expressed as a decimal),
- $ t = 18 $ years,
- $ e $ is the base of the natural logarithm (approximately 2.71828).

Substituting the values into the formula:

\[
V = 62700 \cdot e^{0.026 \times 18}
\]

First, calculate the exponent:

\[
0.026 \times 18 = 0.468
\]

Now, calculate $ e^{0.468} $:

\[
e^{0.468} \approx 1.5978
\]

Finally, multiply by the principal:

\[
V = 62700 \times 1.5978 \approx 100100.06
\]

So, the amount of money in the account after 18 years, to the nearest cent, is **$100,100.06**.

Would you like to explore more details or have any further questions? Here are some related questions:

1. What would the amount be after 25 years with the same interest rate?
2. How would the final amount change if the interest rate were 3% instead?
3. What is the difference in the final amount if the interest is compounded annually instead of continuously?
4. How does the principal affect the final amount for different initial investments?
5. Can you derive the continuous compounding formula from the basic principles of exponential growth?

**Tip:** Continuous compounding generally yields slightly higher returns compared to standard compounding methods due to the nature of exponential growth.

Learn how to calculate the growth of an investment using continuous compounding formula. Explore a detailed example of investing $62,700 at an annual rate of 2.6% over 18 years. Understand the impact of continuous compounding and the natural logarithm base (e) on investment growth.

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