Let's break down the problem step by step.
1. Understanding the Billing Cycle:
- Shirley's credit card billing cycle is 30 days long.
- For the first 2 days, her balance was \[tex]$2830.
- For the remaining \(30 - 2 = 28\) days, her balance was \$[/tex]0 because she paid off her entire balance and didn't make any new purchases.
2. Identifying the Daily Interest Rate:
- The Annual Percentage Rate (APR) is 19%.
- To find the daily interest rate, we divide the APR by the number of days in a year (365 days).
- So, the daily interest rate is [tex]\(\frac{19\%}{365}\)[/tex] or [tex]\(\frac{0.19}{365}\)[/tex].
3. Calculating the Average Daily Balance:
- The average daily balance is calculated as the weighted average of the daily balances during the billing cycle.
- For the first 2 days, the balance was \[tex]$2830, and for the remaining 28 days, the balance was \$[/tex]0.
- The average daily balance [tex]\( \text{(ADB)} \)[/tex] is given by:
[tex]\[
\text{ADB} = \frac{2 \times 2830 + 28 \times 0}{30}
\][/tex]
- Simplifying this, we get:
[tex]\[
\text{ADB} = \frac{2 \times 2830 + 0}{30} = \frac{5660}{30}
\][/tex]
4. Calculating the Interest Charged:
- The interest for the billing cycle is computed based on the daily interest rate, the length of the billing cycle (30 days), and the average daily balance.
- The expression for the interest charged is:
[tex]\[
\left(\frac{0.19}{365} \cdot 30\right) \left(\frac{2 \times 2830 + 28 \times 0}{30}\right)
\][/tex]
Now, we match this to one of the given options.
- Option D: [tex]\(\left(\frac{0.19}{365} \cdot 30\right)\left(\frac{2 \bullet \$ 2830 + 28 \bullet \$ 0}{30}\right)\)[/tex]
This matches the expression we've derived for calculating the interest charge.
Therefore, the correct expression to calculate the amount Shirley was charged in interest for the billing cycle is:
Option D.
