Question

A theater is presenting a program on drinking and driving for students and their parents or other responsible adults. The proceeds will be donated to a local alcohol information center. Admission is $8.00 for adults and $4.00 for students. However, this situation has two constraints: The theater can hold no more than 180 people and for every two adults, there must be at least one student. How

many adults and students should attend to raise the maximum amount of money?
To raise the maximum amount of money. __ adults and __ students should attend.

Answer 1

To raise the maximum amount of money, 120 adults and 60 students should attend.

Explanation:

Let x be the number of adults attending the program.

Let y be the number of students attending the program.

From the given constraints, we can form the following inequalities:

• x + y ≤ 180
• 2y ≥ x
• x ≥ 0
• y ≥ 0

The theater can hold a maximum of 180 people.

For every two adults, there must be at least one student. So the number of students attending the event must be greater than or equal to half the number of adults attending the event.

The number of students and adults must be greater than or equal to zero.

Given the admission is $8.00 for adults and $4.00 for students, the expression for the total amount of money raised, z, is:

z = 8x + 4y

To maximize the value of z, first graph the inequalities and find the feasible region. (The feasible region is the region that is shaded by all of the inequalities).

The corner points of the feasible region are the points of intersection of the boundary lines.

The feasible region is bounded by the corner points:

• (0, 0)
• (0, 180)
• (120, 60)

Evaluate the objective function z = 8x + 4y at each of these corner points:

Point (0, 0): z = 8(0) + 4(0) = 0

Point (0, 180): z = 8(0) + 4(180) = 720

Point (120, 60): z = 8(120) + 4(60) = 1200

Therefore, the maximum value of z is 1200, which occurs at the corner point (120, 60).

Therefore, the maximum amount of money that can be raised is $1,200.

To raise the maximum amount of money, 120 adults and 60 students should attend.