Question

The drama club is selling tickets to their play to raise money for the show's expenses. Each student ticket sells for $6, and each adult ticket sells for $11. The auditorium can hold at most 100 people. The drama club must make no less than $770 from ticket sales to cover the show's costs. If

x represents the number of student tickets sold and
y represents the number of adult tickets sold, write and solve a system of inequalities graphically and determine one possible solution. a. Cost of a student ticket (x): $6
b. Cost of an adult ticket (y): $11
c. Maximum capacity of the auditorium: 100 people
d. Minimum required revenue from ticket sales: $770

Answer 1

To solve the system of inequalities for the drama club's ticket sales, we graph two inequalities representing the maximum capacity of the auditorium (x + y ≤ 100) and the minimum revenue required (6x + 11y ≥ 770). The feasible region where these inequalities overlap gives the possible number of tickets that can be sold. A sample solution is selling 70 student tickets and 30 adult tickets.

Step-by-step explanation:

Solving the System of Inequalities Graphically

To solve the drama club's ticket sales problem, we need to create a system of inequalities based on the given information:

1. The total number of people (x + y) must be no more than 100: x + y ≤ 100.
2. The total revenue (6x + 11y) must be at least $770: 6x + 11y ≥ 770.

Graphically, we plot the two lines corresponding to these equations and shade the areas that meet the given conditions. The feasible region is where both shaded areas overlap, representing all possible combinations of x and y that satisfy both conditions.

Let's graph these inequalities:

1. First, plot the line x + y = 100. This line represents the maximum capacity of the auditorium. Any point on or below this line is acceptable.
2. Second, plot the line 6x + 11y = 770. This line represents the minimum revenue required. Any point on or above this line is acceptable.
3. The intersection of the two shaded regions gives us the feasible solutions.

Here's one possible solution: If the drama club sells 70 student tickets (x = 70) and 30 adult tickets (y = 30), they will reach the full capacity of the auditorium and exceed the minimum revenue requirement since 6(70) + 11(30) = 420 + 330 = $750, which is above $770.

When determining ticket prices and expected revenue, it is important to consider factors such as the demand curve, fixed costs, and audience size. The ideal pricing strategy would balance maximum attendance with maximum revenue while considering the value of the performance to the audience. As costs of production rise, this balance becomes increasingly difficult to maintain.