The problem describes a doctor's schedule and appointment durations. We are given inequalities $0.25 \le y \le 1$, $7 \le x \le 12$, and $xy \le 8$. We are asked to determine what each variable represents and what the expression $xy$ represents in the context of the problem.

1. Problem Description

The problem describes a doctor's schedule and appointment durations. We are given inequalities $0.25 \le y \le 1$, $7 \le x \le 12$, and $xy \le 8$. We are asked to determine what each variable represents and what the expression $xy$ represents in the context of the problem.

2. Solution Steps

a. Determining what each variable represents:

Since the appointment times are 15 minutes, 30 minutes, and 1 hour (60 minutes), and the number of hours the doctor works is no more than 8 hours, it's likely that $y$ represents the appointment time in hours and $x$ represents the number of patients.

From the given information, the appointment times are 15 minutes, 30 minutes, and 1 hour. Converting these to hours gives $\frac{15}{60} = 0.25$, $\frac{30}{60} = 0.5$, and $1$. The inequality $0.25 \le y \le 1$ suggests that $y$ is the duration of an appointment in hours.

The doctor sees between 7 and 12 patients each day. The inequality $7 \le x \le 12$ suggests that $x$ is the number of patients the doctor sees each day.

b. Interpreting the expression $xy$:

Since $x$ represents the number of patients and $y$ represents the duration of each appointment in hours, the expression $xy$ represents the total time spent seeing patients in hours. The inequality $xy \le 8$ means the total time spent seeing patients each day is no more than 8 hours.

3. Final Answer

a. $x$ represents the number of patients the doctor sees each day, and $y$ represents the duration of an appointment in hours.

b. The expression $xy$ represents the total time spent seeing patients each day in hours.