6. A nursing student works at a doctor's office for $15 per hour and tutors other students for $25 per hour. The student cannot work more than 20 hours each week. The student wants to earn at least $375 each week.
• Define two variables and write a system of inequalities that represents the given constraints.
• Suppose the student wants to work at the doctor's office for as many hours as possible for the experience, and suppose the student can only work a whole number of hours at the doctor's office. How many hours should the student work at the doctor's office each week?
Show your work or explain how you found your answer. (4 points)
Let:
x = the number of hours worked at the doctor's office
y = the number of hours worked as a tutor
The system of inequalities can be written as:
x ≥ 0 (since the student cannot work a negative number of hours at the doctor's office)
y ≥ 0 (since the student cannot work a negative number of hours as a tutor)
x + y ≤ 20 (since the student cannot work more than 20 hours each week)
15x + 25y ≥ 375 (since the student wants to earn at least $375 each week)
To maximize the number of hours worked at the doctor's office, we need to find the maximum value of x.
Let's analyze the constraints:
x ≥ 0 (this constraint does not affect the maximum value of x)
x + y ≤ 20 (this constraint limits the sum of x and y to be less than or equal to 20)
15x + 25y ≥ 375 (this constraint ensures that the student earns at least $375 each week)
To find the maximum value of x, we need to find the largest feasible value of x that satisfies all the constraints. The constraint x + y ≤ 20 limits the sum of x and y to be less than or equal to 20. To maximize x, we can set y = 0, which gives x ≤ 20. However, the constraint 15x + 25y ≥ 375 also needs to be satisfied. Substituting y = 0 into this inequality, we get 15x ≥ 375, which simplifies to x ≥ 25. Since x ≤ 20 (due to the constraint x + y ≤ 20) and x ≥ 25 (due to the constraint 15x + 25y ≥ 375 with y = 0), there is no feasible value of x that satisfies both constraints at the same time.
Therefore, we cannot determine the exact number of hours the student should work at the doctor's office each week to maximize the experience, as no feasible solution exists.