To calculate the balance in an account with compound interest, we use the compound interest formula:

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

Where:

• $A$ is the amount of money in the account after the time period.
• $P$ is the principal (initial investment), which is $15,000$.
• $r$ is the annual interest rate (APR), which is 6%, or 0.06 as a decimal.
• $n$ is the number of times the interest is compounded per year. Since the interest is compounded daily, $n = 365$.
• $t$ is the time the money is invested for, which is 6 years.

Now, let's calculate the balance $A$.

$A = 15,000 \left(1 + \frac{0.06}{365}\right)^{365 \times 6}$

I'll compute this for you.The balance in the account after 6 years with daily compounding at an APR of 6% will be approximately $21,499.31.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:

1. How does the balance change if the APR is 8% instead of 6%?
2. What if the investment period was 10 years instead of 6 years?
3. How would the balance differ with monthly compounding instead of daily?
4. What is the difference in balance between simple interest and compound interest over 6 years?
5. How does continuous compounding compare to daily compounding for this investment?

Tip: More frequent compounding periods (like daily or continuously) will typically yield a higher balance over time due to interest being applied more often!