To calculate how much Ethan will have in the account after 6 years, we use the compound interest formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Where:

• $A$ is the final amount (balance in the account)
• $P = 8000$ is the principal (initial amount)
• $r = 0.04$ is the annual interest rate (4% as a decimal)
• $n = 12$ is the number of times the interest is compounded per year (monthly)
• $t = 6$ is the time in years

Now, plug the values into the formula:

$A = 8000 \left(1 + \frac{0.04}{12}\right)^{12 \times 6}$

Let's break this down step by step:

1. Calculate $\frac{0.04}{12} = 0.003333$.
2. Add 1: $1 + 0.003333 = 1.003333$.
3. Multiply the exponent: $12 \times 6 = 72$.
4. Now raise $1.003333$ to the power of 72: $1.003333^{72} \approx 1.27024$
5. Multiply the principal by this value: $A = 8000 \times 1.27024 \approx 10162.00$

Thus, the final balance in the account after 6 years is $10,162.00.

Let me know if you'd like a more detailed breakdown of any step! Here are some related questions to expand your understanding:

1. How would the balance change if the interest rate were higher, say 5% instead of 4%?
2. What happens to the balance if the interest were compounded daily instead of monthly?
3. How would the final amount differ if Ethan invested the money for 10 years instead of 6 years?
4. What is the effect of changing the principal from $8,000 to $10,000?
5. What would the formula look like if the interest were compounded quarterly instead of monthly?

Tip: The more frequently interest is compounded, the more you’ll end up with in the long run!