To determine which expression correctly calculates the interest Jerry was charged for his billing cycle, we need to follow these steps:
1. Identify the billing periods and balances:
- For the first 14 days, Jerry's balance was \[tex]$1050.
- For the last 16 days, Jerry's balance was \$[/tex]1280.
2. Calculate the average daily balance:
The average daily balance is a weighted average based on the number of days each balance was held.
[tex]\[
\text{Average Daily Balance} = \frac{(14 \cdot \$1050) + (16 \cdot \$1280)}{30}
\][/tex]
3. Convert the APR to a daily interest rate:
The APR (Annual Percentage Rate) is 19%. To find the daily interest rate, we divide the APR by 365 (the number of days in a year):
[tex]\[
\text{Daily Interest Rate} = \frac{0.19}{365}
\][/tex]
4. Calculate the interest for the billing cycle:
The interest charged for the billing cycle is the product of the average daily balance, the daily interest rate, and the number of days in the billing cycle (30 days):
[tex]\[
\text{Interest} = \text{Average Daily Balance} \times \text{Daily Interest Rate} \times 30
\][/tex]
Given these calculations, let's break down the options provided:
Option A:
[tex]\[
\left(\frac{0.19}{365} \cdot 30\right) \left(\frac{16 \cdot \$1050 + 14 \cdot \$1280}{30}\right)
\][/tex]
This is incorrect because the balances for the given number of days are switched. Jerry's balance was \[tex]$1050 for 14 days and \$[/tex]1280 for 16 days.
Option B:
[tex]\[
\left(\frac{0.19}{365} \cdot 31\right) \left(\frac{16 \cdot \$1050 + 14 \cdot \$1280}{31}\right)
\][/tex]
This is incorrect for two reasons: the number of days in the billing cycle should be 30 days, not 31, and it incorrectly multiplies the balances with days.
Option C:
[tex]\[
\left(\frac{0.19}{365} \cdot 30\right) \left(\frac{14 \cdot \$1050 + 16 \cdot \$1280}{30}\right)
\][/tex]
This is correct. It accurately uses the daily interest rate over 30 days and correctly weights the balances by the number of days they were held to determine the average daily balance.
Option D:
[tex]\[
\left(\frac{0.19}{365} \cdot 31\right) \left(\frac{14 \cdot \$1050 + 16 \cdot \$1280}{31}\right)
\][/tex]
This is incorrect because the billing cycle is only 30 days, not 31.
Therefore, the correct expression that calculates the interest Jerry was charged in his billing cycle is:
Option C:
[tex]\[
\left(\frac{0.19}{365} \cdot 30\right)\left(\frac{14 \cdot \$1050 + 16 \cdot \$1280}{30}\right)
\][/tex]
