Problem 36 - Precalculus: Functions and Graphs

Question

An auditorium contains 600 seats. For an upcoming event, tickets will be priced at 8 dollars for some seats and 5 dollars for others. At least 225 tickets are to be priced at 5 dollars, and total sales of at least 3000 are desired. Find and graph a system of inequalities that describes all possibilities for pricing the two types of tickets.

Step-by-Step Solution

Answer

The inequalities are \(x+y\leq600\), \(y\geq225\), \(8x+5y\geq3000\), \(x\geq0\), \(y\geq0\).

Step 1: Define Variables

Let \(x\) represent the number of tickets sold at \\(8 each, and \(y\) represent the number of tickets sold at \\)5 each.

Step 2: Set Up Inequality for Total Seats

The auditorium contains 600 seats, so we have the inequality \(x + y \leq 600\).

Step 3: Set Up Inequality for 5 Dollar Tickets

At least 225 tickets must be sold at \$5 each, so we have \(y \geq 225\).

Step 4: Set Up Inequality for Total Sales

The total sales must be at least \$3000. Therefore, the inequality for this is \(8x + 5y \geq 3000\).

Step 5: Additional Constraints

Tickets must have non-negative values, so we include \(x \geq 0\) and \(y \geq 0\).

Step 6: Graph the System of Inequalities

Plot each inequality on a coordinate plane. Shade the feasible region that satisfies all the inequalities: \(x + y \leq 600\), \(y \geq 225\), \(8x + 5y \geq 3000\), \(x \geq 0\), and \(y \geq 0\). Determine where these regions overlap.

Key Concepts

Linear Programming

Linear programming is a mathematical approach used to achieve the best outcome in a mathematical model. It's specifically designed for scenarios with certain constraints, like maximizing profits or minimizing costs.

It involves several components:

• Objective Function: This is the formula we want to optimize, whether it's for maximum or minimum value. In our exercise, we want to maximize the total sales by adjusting the number of $8 and $5 tickets sold.
• Variables: These represent the quantities we want to find. Here, 'x' is the number of $8 tickets sold, and 'y' is for $5 tickets.
• Constraints: These are the rules or limits within which our solution must lie. In this problem, constraints include the total number of seats, a minimum number of $5 tickets, and a sales revenue requirement.

By finding a solution that satisfies all constraints and optimizes the objective function, linear programming provides the most efficient outcome. In practice, this involves graphing the system of inequalities to identify feasible solutions.

Graphing Inequalities

Graphing inequalities is a visual method for solving systems of inequalities. It helps us understand all possible solutions that meet the defined constraints by sketching them on a graph.

To graph an inequality:

• Start with the equation form of the inequality. For example, for the inequality \(x + y \leq 600\), graph the line \(x + y = 600\).
• Choose a point not on the line (like (0,0)) to test which side of the line satisfies the inequality. If the chosen point satisfies the inequality, shade that side of the line. Otherwise, shade the opposite side.
• Repeat for each inequality in the system, such as \(y \geq 225\) or \(8x + 5y \geq 3000\).
• The overlapping shaded region represents all possible solutions (feasible region) for the system of inequalities.

This approach gives a clear picture of where all conditions intersect, pointing out feasible solutions for the pricing and number of tickets sold at different price points.

Constraints in Math

Constraints in mathematical problems are limitations or conditions that the solution must satisfy. They are essential for describing real world situations realistically.

In our context, these constraints govern how we can distribute tickets and make sales:

• Capacity Constraint: The total seats available set a limit. We express this as \(x + y \leq 600\), meaning the number of tickets sold cannot exceed available seats.
• Specific Minimums: Such as having at least 225 tickets sold at \(5. This is represented as \(y \geq 225\).
• Total Revenue Requirement: This ensures sales reach a minimum of \)3000, denoted as \(8x + 5y \geq 3000\).
• Non-negativity: We assume number of tickets sold cannot be negative, hence \(x \geq 0\) and \(y \geq 0\).

These constraints define the solution space and help narrow down feasible solutions for optimum decision making. Understanding these boundaries makes it easier to address real-life scenarios with precision.