Question: 3 — Continuously Compounding Interest
LEARNING OBJECTIVE: Solve for account balance when interest is compounded continuously.
If $2,500$ is invested at $12\%$ annual interest, which is compounded continuously, what is the account balance after $3$ years, assuming no additional deposits or withdrawals are made?
a.) $3,683.46$
b.) $3,512.32$
c.) $3,489.58$
d.) $3,583.32
1. **State the problem:** We want to find the account balance after $3$ years when $2,500$ is invested at an annual interest rate of $12\%$ compounded continuously.
2. **Formula for continuous compounding:** The formula to calculate the amount $A$ after time $t$ years with principal $P$ and annual interest rate $r$ compounded continuously is:
$$A = P e^{rt}$$
where $e$ is Euler's number (approximately $2.71828$).
3. **Identify values:**
- Principal $P = 2500$
- Rate $r = 0.12$ (since $12\% = 0.12$)
- Time $t = 3$ years
4. **Substitute values into the formula:**
$$A = 2500 \times e^{0.12 \times 3} = 2500 \times e^{0.36}$$
5. **Calculate the exponent:**
$$e^{0.36} \approx 1.433329$$
6. **Calculate the amount:**
$$A = 2500 \times 1.433329 = 3583.3225$$
7. **Round to two decimal places:**
$$A \approx 3583.32$$
**Final answer:** The account balance after $3$ years is approximately $3,583.32$, which corresponds to option d.)