Sure! Let's solve this problem step by step to determine which of the given expressions is correct.
1. Determine the average daily balance:
a. For the first 12 days, Sergei's balance was \$350. Therefore, the total balance for these days is:
[tex]\[
350 \times 12 = 4200
\][/tex]
b. For the last 18 days, his balance was \$520. Therefore, the total balance for these days is:
[tex]\[
520 \times 18 = 9360
\][/tex]
c. The total balance over the entire billing cycle is the sum of the balances above:
[tex]\[
4200 + 9360 = 13560
\][/tex]
d. The number of days in the billing cycle is 30 days. Therefore, the average daily balance is:
[tex]\[
\text{Average daily balance} = \frac{13560}{30} = 452 \text{ dollars}
\][/tex]
2. Determine the daily periodic rate:
The Annual Percentage Rate (APR) is 14%. To find the daily periodic rate, divide the APR by the number of days in a year (365):
[tex]\[
\text{Daily periodic rate} = \frac{0.14}{365} \approx 0.0003835616438356165
\][/tex]
3. Calculate the amount of interest charged:
The billing cycle has 30 days, so the interest charged can be found by multiplying the daily periodic rate by the number of days in the billing cycle and by the average daily balance:
[tex]\[
\text{Interest charged} = 0.0003835616438356165 \times 30 \times 452 \approx 5.20109589041096 \text{ dollars}
\][/tex]
4. Identify the correct expression:
Given the structure of the options, the correct expression is of the form:
[tex]\[
\left(\frac{0.14}{365} \cdot 30\right)\left(\frac{12 \cdot 350 + 18 \cdot 520}{30}\right)
\][/tex]
This matches option B:
[tex]\[
\left(\frac{0.14}{365} \cdot 30\right)\left(\frac{12 \cdot 350 + 18 \cdot 520}{30}\right)
\][/tex]
Thus, the correct expression to calculate the amount Sergei was charged in interest for the billing cycle is:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{12 \cdot 350 + 18 \cdot 520}{30}\right) \][/tex]
