Certainly! Let's break down this problem step-by-step.
### Step 1: Understand the Balances and Days
Trina's balance changes occur as follows:
1. For the first 10 days, her balance was \$780.
2. After making a purchase of \[tex]$170, her balance becomes \$[/tex]950 for the next 10 days.
3. After making a payment of \[tex]$210, her balance for the last 10 days was \$[/tex]740.
### Step 2: Calculate the Average Daily Balance
To calculate the average daily balance, we'll use the formula:
[tex]\[
\text{Average Daily Balance} = \frac{\sum (\text{Balance on each day})}{\text{Number of days}}
\][/tex]
Given the balance details:
- First 10 days: \$780
- Next 10 days: \$950
- Last 10 days: \$740
We calculate the total balance over the 30 days as follows:
[tex]\[
\text{Total Balance} = (10 \times 780) + (10 \times 950) + (10 \times 740)
\][/tex]
Now compute the average daily balance:
[tex]\[
\text{Average Daily Balance} = \frac{(10 \times 780) + (10 \times 950) + (10 \times 740)}{30}
\][/tex]
[tex]\[
= \frac{7800 + 9500 + 7400}{30}
\][/tex]
[tex]\[
= \frac{24700}{30}
\][/tex]
The average daily balance is:
[tex]\[
= 823.3333333333334
\][/tex]
### Step 3: Calculate the Daily Interest Rate
The Annual Percentage Rate (APR) is 17%, and we need to find the daily interest rate:
[tex]\[
\text{Daily Interest Rate} = \frac{APR}{365} = \frac{0.17}{365}
\][/tex]
The daily interest rate is:
[tex]\[
= 0.0004657534246575343
\][/tex]
### Step 4: Calculate the Interest Charged for the Billing Cycle
Using the average daily balance and the daily interest rate, we can calculate the interest for the 30-day cycle:
[tex]\[
\text{Interest} = (\text{Daily Interest Rate}) \times (\text{Number of days in billing cycle}) \times (\text{Average Daily Balance})
\][/tex]
[tex]\[
= 0.0004657534246575343 \times 30 \times 823.3333333333334
\][/tex]
The amount of interest charged is:
[tex]\[
= 11.504109589041098
\][/tex]
### Step 5: Identify the Correct Expression
We have our calculated interest amount, and we need to match it with the given options. The correct expression that matches our results is:
[tex]\[
\left( \frac{0.17}{365} \cdot 30 \right) \left( \frac{10 \cdot 780 + 10 \cdot 950 + 10 \cdot 740}{30} \right)
\][/tex]
This is equivalent to:
[tex]\[
\left( \frac{0.17}{365} \cdot 30 \right) \left( 823.3333333333334 \right)
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\left(\frac{0.17}{365} \cdot 30\right)\left(\frac{10 \cdot 780+10 \cdot 950+10 \cdot 740}{30}\right)}
\][/tex]
This is the correct formula that can be used to calculate the amount of interest Trina was charged during the billing cycle.
