Question: Suppose $24,000$ is deposited into an account paying $7.25\%$ interest, which is compounded continuously.
How much money will be in the account after ten years if no withdrawals or additional deposits are made?
$47,897.10$
$46,414.20$
$48,326.40$
$49,553.54$
1. **State the problem:** We want to find the amount of money in an account after $10$ years with continuous compounding interest.
2. **Formula for continuous compounding:** The amount $A$ after time $t$ years is given by:
$$A = P e^{rt}$$
where:
- $P$ is the principal (initial deposit),
- $r$ is the annual interest rate (as a decimal),
- $t$ is the time in years,
- $e$ is Euler's number, approximately $2.71828$.
3. **Identify values:**
- $P = 24000$
- $r = 7.25\% = 0.0725$
- $t = 10$
4. **Calculate the exponent:**
$$rt = 0.0725 \times 10 = 0.725$$
5. **Calculate $A$:**
$$A = 24000 \times e^{0.725}$$
6. **Evaluate $e^{0.725}$:**
Using a calculator, $e^{0.725} \approx 2.0653$
7. **Multiply:**
$$A = 24000 \times 2.0653 = 49567.2$$
8. **Rounding:**
The amount after $10$ years is approximately $49567.20$.
9. **Compare with given options:**
The closest option is $49,553.54$.
**Final answer:** $49,553.54$