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 \( \$ 12.00 \) for adults and \( \$ 6.00 \) for students. However, this situation has two constraints: The theater can hold no more than 210 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.

Step 1 ：Let x be the number of adults and y be the number of students.

Step 2 ：Set up the constraint equations: \(x+y \le 210\), \(2y \le x\).

Step 3 ：Set up the objective function, which represents the total proceeds: \(P(x,y) = 12x + 6y\).

Step 4 ：Solve the inequality \(2y \le x\) to find the relationship between x and y: \(y \ge \frac{1}{2}x\).

Step 5 ：Since the number of participants cannot exceed 210, we have \(x + y = 210\). Substitute the previous relationship: \(x + \frac{1}{2}x = 210\).

Step 6 ：Solve the equation for x: \(\frac{3}{2}x = 210\) => \(x = 140\).

Step 7 ：Find y using the equation \(x + y = 210\) => \(y = 70\).

Step 8 ：The maximum proceeds can be obtained with 140 adults and 70 students attending.