Question: Suppose $30,000$ is deposited into an account paying $3.5\%$ interest, compounded annually.
How much money is in the account after ten years if no withdrawals or additional deposits are made?
1. **State the problem:** We want to find the amount of money in an account after $10$ years with an initial deposit of $30,000$ and an annual interest rate of $3.5\%$, compounded annually.
2. **Formula used:** The compound interest formula is $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ where:
- $A$ is the amount of money accumulated after $t$ years, including interest.
- $P$ is the principal amount (initial deposit).
- $r$ is the annual interest rate (decimal).
- $n$ is the number of times interest is compounded per year.
- $t$ is the number of years.
3. **Identify values:**
- $P = 30000$
- $r = 0.035$ (since $3.5\% = 0.035$)
- $n = 1$ (compounded annually)
- $t = 10$
4. **Substitute values into the formula:**
$$A = 30000 \left(1 + \frac{0.035}{1}\right)^{1 \times 10} = 30000 \left(1 + 0.035\right)^{10} = 30000 \times 1.035^{10}$$
5. **Calculate $1.035^{10}$:**
$$1.035^{10} \approx 1.4106$$
6. **Calculate the amount $A$:**
$$A = 30000 \times 1.4106 = 42317.96$$
7. **Interpretation:** After $10$ years, the account will have approximately $42,317.96$.
**Final answer:** $42,317.96$