Final answer:
An inequality to represent the number of 10-minute (x) and 20-minute (y) doctor's appointments per week, given a four-hour paperwork commitment each weekday and a maximum of 30 work hours per week, is 3x + 2y ≤ 7200.
Step-by-step explanation:
If a doctor's office schedules 10-minute and 20-minute appointments and completes paperwork for four hours each weekday (Monday to Friday), the doctor wishes to limit these activities to at most 30 hours per week. Let's define x as the number of 10-minute appointments and y as the number of 20-minute appointments she may have in a week.
To find the inequality that represents the number of each type of office visit she may have in a week, we need to consider the time spent on these activities. The doctor works 5 days a week and spends 4 hours a day on paperwork, for a total of 20 hours on paperwork weekly. This leaves 10 hours for appointments, since the doctor wants to work at most 30 hours per week.
A 10-minute appointment is ⅔ of an hour, and a 20-minute appointment is ⅓ of an hour. So the total time, T, spent on appointments can be expressed as:
T = ⅔x + ⅓y
This time should not exceed the remaining 10 hours (600 minutes) available after paperwork:
⅔x + ⅓y ≤ 600 minutes
To express this time constraint as an inequality, we multiply the entire inequality by 12 to get rid of fractions:
(3x) + (2y) ≤ 7200
Therefore, the inequality representing the maximum number of each type of appointment is:
3x + 2y ≤ 7200