Question: Calculate the total value of the investment given the following information and compounded continuously.
Principal $1000; rate 4.5%; term 10 years
Select the correct response:
$1568.31
$1568.82
$1568.27
$1568.57
1. **State the problem:** Calculate the total value of an investment compounded continuously with principal $1000$, rate $4.5\%$, and term $10$ years.
2. **Formula used:** The formula for continuous compounding is $$A = Pe^{rt}$$ where:
- $A$ is the amount after time $t$
- $P$ is the principal
- $r$ is the annual interest rate (as a decimal)
- $t$ is the time in years
3. **Convert rate to decimal:**
$$r = 4.5\% = 0.045$$
4. **Substitute values:**
$$A = 1000 \times e^{0.045 \times 10}$$
5. **Calculate exponent:**
$$0.045 \times 10 = 0.45$$
6. **Evaluate exponential:**
$$e^{0.45} \approx 1.568312185$$
7. **Calculate amount:**
$$A = 1000 \times 1.568312185 = 1568.312185$$
8. **Round to two decimal places:**
$$A \approx 1568.31$$
**Final answer:** The total value of the investment is **1568.31**.