To determine what Stephanie's credit card balance will be at the end of 12 months with daily compounding interest, we need to use the compound interest formula. Below is a step-by-step explanation of how we can calculate it:
1. Identify the given values:
- Initial balance ([tex]\( P \)[/tex]): [tex]$11,300
- Annual Percentage Rate (APR or \( r \)): 12% or 0.12
- Number of compounding periods per year ( \( n \)): 365 (since interest is compounded daily)
- Time in years (\( t \)): 1 year
2. Formula for compound interest:
The compound interest formula is given by:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial balance).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for in years.
3. Substitute the values into the formula:
\[
A = 11300 \left(1 + \frac{0.12}{365}\right)^{365 \times 1}
\]
4. Calculate the daily interest rate:
\[
\frac{0.12}{365} \approx 0.000328767
\]
5. Calculate the exponent component:
\[
365 \times 1 = 365
\]
6. Plug the values back into the compound interest formula:
\[
A = 11300 \left(1 + 0.000328767\right)^{365}
\]
7. Calculate the final amount:
Using a calculator to compute the above expression, we get:
\[
A \approx 12740.46
\]
Thus, at the end of 12 months, Stephanie's credit card balance will be approximately $[/tex]12,740.46. Therefore, the correct answer is:
B. $12,740.46
