To determine the amount Marlene was charged in interest for her billing cycle, we need to follow these detailed steps:
1. Determine the balances for different periods of the billing cycle:
- For the first 10 days, Marlene's balance was \[tex]$570.
- For the next 10 days, after making a purchase of \$[/tex]120, her balance increased to \[tex]$690.
- For the final 10 days, after making a payment of \$[/tex]250, her balance decreased to \$440.
2. Calculate the average daily balance (ADB):
The average daily balance is calculated by averaging the balances over the various periods in the billing cycle. Since each period is 10 days long in a 30-day billing cycle:
[tex]\[
\text{Average Daily Balance} = \frac{(10 \times 570) + (10 \times 690) + (10 \times 440)}{30}
\][/tex]
Evaluating this:
[tex]\[
\text{Average Daily Balance} = \frac{5700 + 6900 + 4400}{30} = \frac{17000}{30} = 566.6666666666666
\][/tex]
3. Calculate the daily periodic rate (DPR):
The APR (Annual Percentage Rate) is 15%, which is divided by the number of days in a year (365) to find the daily periodic rate:
[tex]\[
\text{DPR} = \frac{0.15}{365} \approx 0.000410958904109589
\][/tex]
4. Calculate the interest for the billing cycle:
To find the interest charged for the 30-day billing cycle, multiply the daily periodic rate by the number of days in the billing cycle and the average daily balance:
[tex]\[
\text{Interest} = \left(\frac{0.15}{365} \cdot 30\right) \left(566.6666666666666\right)
\][/tex]
Evaluating this:
[tex]\[
\text{Interest} \approx 0.000410958904109589 \times 30 \times 566.6666666666666 \approx 6.986301369863013
\][/tex]
Considering the given expressions, let's see which one matches our findings:
- Option A: [tex]\(\left(\frac{0.15}{365} \cdot 30\right)(5570)\)[/tex]
- Option B: [tex]\(\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot 570+10 \cdot 690+10 \cdot 440}{30}\right)\)[/tex]
- Option C: [tex]\(\left(\frac{0.15}{365} \cdot 30\right)(5320)\)[/tex]
Clearly, option B correctly describes the calculation of the average daily balance and the interest charge:
[tex]\[
\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot 570 + 10 \cdot 690 + 10 \cdot 440}{30}\right)
\][/tex]
Therefore, the correct expression to calculate the amount Marlene was charged in interest for the billing cycle is:
[tex]\[
\boxed{B}
\][/tex]
