Final answer:
The family's rebate at the end of the year would be $116.
Step-by-step explanation:
To calculate the family's rebate at the end of the year, we need to multiply their total gasoline purchases by the cash-back rebate percentage. In this case, the family spent $5800 on gasoline purchases. The cash-back rebate is 2% of the total amount spent. Therefore, the rebate amount would be $5800 multiplied by 2%, which is $116.