Sure, let's break down the problem step-by-step to find which expression correctly calculates the amount Marlene was charged in interest for the billing cycle.
### Step 1: Identify periods and balances
- For the first 10 days, her balance was [tex]$570.
- She made a purchase of $[/tex]120, so her balance for the next 10 days was [tex]$690 ($[/tex]570 + [tex]$120).
- Then, she made a payment of $[/tex]250, so her balance for the last 10 days of the billing cycle was [tex]$440 ($[/tex]690 - [tex]$250).
### Step 2: Calculate the average daily balance
The average daily balance can be found by multiplying each balance by the number of days it was held, summing these amounts, and then dividing by the total number of days in the billing cycle.
\[
\text{Average Daily Balance} = \frac{(10 \cdot 570) + (10 \cdot 690) + (10 \cdot 440)}{30}
\]
Calculating each term:
- \(10 \cdot 570 = 5700\)
- \(10 \cdot 690 = 6900\)
- \(10 \cdot 440 = 4400\)
Sum these values:
\[
5700 + 6900 + 4400 = 17000
\]
Now, divide by the total number of days in the billing cycle (30):
\[
\text{Average Daily Balance} = \frac{17000}{30} \approx 566.6666666666666
\]
### Step 3: Calculate the interest charged
The interest is calculated using the formula:
\[
\text{Interest Charged} = \left(\frac{{\text{APR}}}{\text{Days in Year}} \cdot \text{Billing Cycle Days}\right) \cdot \text{Average Daily Balance}
\]
Plugging in the given values (APR = 0.15, Days in Year = 365, Billing Cycle Days = 30):
\[
\text{Interest Charged} = \left(\frac{0.15}{365} \cdot 30\right) \cdot 566.6666666666666
\]
Therefore, the correct answer to which expression calculates the amount Marlene was charged in interest for the billing cycle is:
\[
\boxed{\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot \$[/tex] 570+10 \cdot \[tex]$ 690+10 \cdot \$[/tex] 440}{30}\right)}
\]
So, option C is correct.
