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 coly 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.
Let x be the number of hours the student works at the doctor's office and y be the number of hours the student tutors.
The system of inequalities representing the given constraints is:
x + y ≤ 20 (cannot work more than 20 hours each week)
15x + 25y ≥ 375 (wants to earn at least $375 each week)
To find how many hours the student should work at the doctor's office each week, we can use the fact that the student wants to work at the doctor's office for as many hours as possible. This means that the student should work at the doctor's office for all the hours remaining after tutoring for the minimum amount needed to earn $375.
Let's find the minimum amount of hours the student needs to tutor each week to earn $375. Using the second inequality:
15x + 25y ≥ 375
25y ≥ 375 - 15x
y ≥ (375 - 15x)/25
Since y has to be a whole number, the smallest value of y that satisfies this inequality is y = 15 (when x = 0). Therefore, the student needs to tutor for at least 15 hours each week to earn $375.
The maximum number of hours the student can work at the doctor's office is then:
x = 20 - y
x = 20 - 15
x = 5
So the student should work 5 hours at the doctor's office each week and tutor for 15 hours to earn at least $375.
Let's denote:
x = the number of hours the student works at the doctor's office each week
y = the number of hours the student tutors other students each week
Given the constraints:
1. The student cannot work more than 20 hours each week:
x + y ≤ 20
2. The student wants to earn at least $375 each week:
15x + 25y ≥ 375
To determine how many hours the student should work at the doctor's office each week, we need to find the maximum value of x within the given constraints.
One way to find the maximum value of x is to graph the system of inequalities on a coordinate plane and find the feasible region. However, since it is mentioned that the student can only work a whole number of hours at the doctor's office, we can try substituting different values for x and check if it satisfies the given constraints.
Let's substitute x = 1 into the first constraint:
1 + y ≤ 20
y ≤ 19
Substituting x = 1 into the second constraint:
15(1) + 25y ≥ 375
25y ≥ 375 - 15
25y ≥ 360
y ≥ 14.4
Since y must also be a whole number, the maximum value for y is 14. Therefore, the student can work a maximum of 1 hour at the doctor's office each week.