Question: Ben opened a savings account and deposited $3,519.00 as principal. The account earns 15% interest, compounded annually. What is the balance after 7 years? Use the formula $$A=P\left(1+\frac{r}{n}\right)^{nt}$$, where $A$ is the balance (final amount), $P$ is the principal (starting amount), $r$ is the interest rate expressed as a decimal, $n$ is the number of times per year that the interest is compounded, and $t$ is the time in years. Round your answer to the nearest cent.
1. **State the problem:**
Ben deposited $3519$ as principal in a savings account with an interest rate of $15\%$ compounded annually. We want to find the balance after $7$ years.
2. **Write the formula:**
The compound interest formula is:
$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
where:
- $A$ is the final amount (balance),
- $P$ is the principal,
- $r$ is the annual interest rate as a decimal,
- $n$ is the number of times interest is compounded per year,
- $t$ is the number of years.
3. **Identify values:**
- $P = 3519$
- $r = 0.15$ (since $15\% = 0.15$)
- $n = 1$ (compounded annually)
- $t = 7$
4. **Substitute values into the formula:**
$$A = 3519 \left(1 + \frac{0.15}{1}\right)^{1 \times 7} = 3519 \left(1 + 0.15\right)^7 = 3519 \times 1.15^7$$
5. **Calculate $1.15^7$:**
$$1.15^7 = 2.75903125$$
6. **Calculate the final amount:**
$$A = 3519 \times 2.75903125 = 9707.27$$
7. **Round to the nearest cent:**
The balance after 7 years is approximately $9707.27$.
**Final answer:**
$$\boxed{9707.27}$$