To calculate the amount of money in an account after a certain period with continuous compounding interest, we use the formula for continuous compounding:

$A = P e^{rt}$

where:

• $A$ is the amount of money in the account after time $t$.
• $P$ is the principal amount (initial deposit).
• $r$ 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:

• $P = 2000$ dollars
• $r = 0.05$ (5% interest rate)
• $t = 10$ years

We substitute these values into the formula:

$A = 2000 \cdot e^{0.05 \cdot 10}$

First, calculate the exponent:

$0.05 \cdot 10 = 0.5$

Next, find $e^{0.5}$:

$e^{0.5} \approx 1.64872$

Now, multiply this by the principal amount:

$A = 2000 \cdot 1.64872$ $A \approx 3297.44$

Therefore, the amount of money in the account after 10 years will be approximately $3,297.44.

Would you like further details or have any questions about this solution?

Here are 5 more questions you might want to ask:

1. How do you calculate compound interest compounded annually?
2. What is the difference between simple interest and compound interest?
3. How do you solve exponential growth and decay problems?
4. Can you explain the concept of present value in finance?
5. How does the Rule of 72 help estimate the doubling time of an investment?

Tip: Continuous compounding grows investments at a faster rate than simple or periodic compounding because the interest is constantly being added to the principal balance.