Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Ticket prices An auditorium contains 600 seats. For an upcoming event, tickets will be priced at $$ 8\( for some seats and $$ 5\) for others. At least 225 tickets are to be priced at $$ 5\(, 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 system of inequalities is: \( x + y \leq 600 \), \( y \geq 225 \), \( 8x + 5y \geq 3000 \).

Step 1: Define Variables

Let \( x \) be the number of \(8 tickets, and \( y \) be the number of \)5 tickets. These are the variables we need to use to form our inequalities.

Step 2: Total Seats Inequality

The auditorium contains 600 seats, so the sum of tickets sold must be less than or equal to 600. This is expressed by the inequality: \[ x + y \leq 600 \]

Step 3: Minimum $5 Tickets Inequality

At least 225 tickets must be priced at $5, leading to the inequality: \[ y \geq 225 \]

Step 4: Total Sales Inequality

The total sales must be at least \(3000. If \( x \) tickets are sold at \)8 and \( y \) tickets are sold at $5, the total sales can be expressed as \( 8x + 5y \). Therefore, we have the inequality: \[ 8x + 5y \geq 3000 \]

Step 5: Graph Inequalities

Graph each of the inequalities on the coordinate plane with \( x \) on the horizontal axis and \( y \) on the vertical axis. 1. Plot the line \( x + y = 600 \) and shade below it to represent \( x + y \leq 600 \).2. Plot the line \( y = 225 \) and shade above it to represent \( y \geq 225 \).3. Plot the line \( 8x + 5y = 3000 \) and shade above it to represent \( 8x + 5y \geq 3000 \).The overlapping shaded region represents all the possible solutions.

Key Concepts

Linear Programming

Linear programming is a powerful mathematical technique used to find the best outcome in a given mathematical model. In simpler terms, it's about optimizing a certain objective, such as maximizing profit or minimizing cost. In our exercise, we are trying to find the best mix of ticket prices to meet certain conditions. This involves:

• Identifying the variables and constraints, such as the number of seats and ticket prices.
• Setting up inequalities that model these constraints.
• Graphing these inequalities to find a region of feasible solutions.

The objective could be either maximizing profit or ensuring a minimum requirement is met, like a specific amount of total sales, which is expressed in the form of inequalities. A common method to solve these models is to find where the constraints overlap, representing the feasible region where all conditions are met.

Graphing Inequalities

Graphing inequalities is an essential step in solving a system of inequalities. This visual method helps to easily identify the region where all conditions or constraints overlap. To graph an inequality:

• First, rearrange the inequality equation as needed.
• Plot the boundary line. For example, if the inequality is in the form of \( ax + by \leq c \), first graph the equation \( ax + by = c \).
• Determine the area to shade: Use a test point to decide which side of the line represents the inequality.

In our ticket price exercise, this involves plotting three key lines that represent the constraints on total seats, minimum priced tickets, and total sales. By shading the appropriate regions for all inequalities and finding the intersection, we determine the area of possible solutions.

Variables in Inequalities

Variables in inequalities are the unknowns we solve for, much like how we solve for \( x \) in algebra. In our ticket pricing scenario, the variables are:

• \( x \): the number of \(8 tickets
• \( y \): the number of \)5 tickets

These variables are tied to specific conditions that need to be satisfied, forming our inequalities. It's crucial to define these variables properly because they dictate what each inequality represents. For instance, \( x + y \leq 600 \) ensures that the sum of all tickets doesn't exceed the auditorium's capacity. Meanwhile, \( y \geq 225 \) guarantees a minimum number of cheaper tickets sold, and \( 8x + 5y \geq 3000 \) ensures that the sales goal is met. Correctly defining and interpreting these variables is key to analyzing and solving the system of inequalities.