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Ariana invested $7,500 in an account paying an interest rate of 2.9% compounded continuously. Assuming no deposits or withdrawals are made, how much money, *to the nearest ten dollars*, would be in the account after 14 years?

Answer

Attempt 1 out of 2

To solve this problem, we'll use the formula for continuous compound interest, which is given by:

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

Where:
- $ A $ is the amount of money in the account after time $ t $.
- $ P $ is the principal amount (initial investment), which is $7,500.
- $ r $ is the annual interest rate (expressed as a decimal), which is 2.9%, or 0.029.
- $ t $ is the time the money is invested for, in years, which is 14 years.
- $ e $ is the base of the natural logarithm, approximately equal to 2.71828.

### Step-by-Step Calculation:
1. **Substitute the given values into the formula**:
   \[
   A = 7500 \cdot e^{0.029 \times 14}
   \]

2. **Calculate the exponent**:
   \[
   0.029 \times 14 = 0.406
   \]

3. **Calculate the exponential factor**:
   \[
   e^{0.406} \approx 1.5002
   \]

4. **Final Calculation**:
   \[
   A = 7500 \times 1.5002 \approx 11251.5
   \]

### Rounding to the Nearest Ten Dollars:
\[
A \approx 11250 \text{ dollars}
\]

So, the amount in the account after 14 years would be **$11,250**.

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Would you like more details or have any questions? Here are 5 related questions you might consider:

1. How does continuous compounding differ from other compounding methods?
2. What would the final amount be if the interest rate was 4% instead of 2.9%?
3. How would the final amount change if the investment period was only 10 years?
4. What is the effect of increasing the principal amount on the final balance?
5. How much would Ariana need to invest initially to have $15,000 after 14 years with the same interest rate?

**Tip:** Continuous compounding is often used in financial mathematics because it represents the theoretical limit of compounding frequency, leading to the highest possible interest earnings.

Explore how to calculate the amount of money in an account after 14 years with continuous compound interest. This problem involves using the formula A = P \cdot e^{rt}, where A is the final amount, P is the principal, r is the interest rate, and t is the time in years. Get a detailed step-by-step solution and understand the application of continuous compounding. Suitable for high school students.

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