Problem 33 - Precalculus with Limits

Question

Describing an Unusual Characteristic, the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and where they occur. $$ \begin{array}{c}{\text { Objective function: }} \\ {z=3 x+4 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+y \leq 1} \\ {2 x+y \leq 4}\end{array} $$

Step-by-Step Solution

Answer

The unusual characteristic is that the problem's solution region is a bounded triangle. The minimum value of the objective function is 0 which occurs at the point (0,0). And the maximum value of the objective function is 4 and it occurs at the point (0,1).

Step 1: Graphing the Constraints

Start by graphing the constraints using a Cartesian plane. The constraint \(x \geq 0\) means that \(x\) is nonnegative and will be graphed right to the y-axis. Similarly, \(y \geq 0\) means \(y\) is nonnegative and will be graphed above the x-axis. The constraint \(x + y \leq 1\) will divide the plane into two halves with a straight line passing through (0,1) and (1,0) and the solution would lie on or below this line. Lastly, the constraint \(2x + y \leq 4\) can also be plotted on the graph by first finding points that satisfy the equation like: (0,4) and (2,0) and the solution would be on or below this line too.

Step 2: The Unusual Characteristic

Now, looking at the graph, it can be seen that the last two constraints intersect at (0.66,0.33), which is in the first quadrant. But the solution region is an unusual shape - a triangle, which is the region satisfying all the constraints. In this problem, the solution region is not the usual unbounded region, instead, it's bounded, it's a triangle in the first quadrant. This is the unusual characteristic of this problem.

Step 3: The Objective Function

Now, let's find the maximum and minimum values of the objective function \(z = 3x + 4y\). This is done by using the vertices of the bounded region. These are (0,0), (1,0) and (0,1). Substituting in we get z values of 0, 3 and 4 respectively. So the minimum value of the objective function is 0 at (0,0) and the maximum value is 4 at (0,1).

Key Concepts

Objective Function

In linear programming, the goal is to either maximize or minimize a particular quantity, which is represented by the objective function. The objective function is a mathematical expression you want to evaluate depending on different variables. In our exercise, the objective function is given by \[ z = 3x + 4y \]

• The coefficient of each variable in the function indicates its impact on the objective value. Here, every unit increase of \(x\) contributes 3 units to \(z\) and every unit increase of \(y\) adds 4 units.
• The primary aim is to find the highest or the lowest possible value of \(z\), considering the imposed constraints.

This involves evaluating the objective function at the corners of the feasible region, which will be detailed later. Understanding the objective function is crucial as it directly associates with what you aim to optimize in linear programming.

Constraints

Constraints in linear programming are conditions that the solution must satisfy. They define the limits within which the objective function needs to be optimized. In this problem, the constraints are:

• \( x \geq 0 \): This ensures that \(x\) is nonnegative and only takes values on or to the right of the y-axis.
• \( y \geq 0 \): Ensures nonnegative values of \(y\), meaning the values are on or above the x-axis.
• \( x + y \leq 1 \): This forms a boundary line on the plane, with the feasible region being below or on this line.
• \( 2x + y \leq 4 \): Another boundary that defines a line, with the permissible solutions lying on or below it.

Constraints are essentially the rules of the problem. They help in narrowing down the possibilities to find an optimal solution within a feasible area.

Solution Region

The solution region, also known as the feasible region, is the set of all points that satisfy the constraints. When you graph the constraints, the solution region is where all the conditions intersect. In our problem, this takes the form of a triangle in the first quadrant.

• This region is bounded because it has a limited area and is fully enclosed by the constraints' lines.
• The vertices of the triangular region are critical; they provide points for evaluating the objective function.

The solution region is crucial because only within this area can the objective function be optimized. Understanding its shape and boundaries helps determine feasible points to test for maximum and minimum values.

Graphing Constraints

Graphing constraints is an essential step in visualizing the linear programming problems. Each constraint describes a boundary that restricts the possible solutions.

• Start by identifying each equation or inequality. Here, for example, \( x + y \leq 1 \) and \( 2x + y \leq 4 \) are plotted as lines.
• Determine key points where these lines intersect the axes, like (0,1) and (1,0) for \( x + y = 1 \).
• Shade the feasible region, which is the intersection of all constraint-driven subregions.

By graphing these constraints on a Cartesian plane, it becomes clear how they divide the space and where optimal solutions could reside. Properly graphing constraints guides the finding of feasible solutions effectively.