Problem 16 - Precalculus with Limits

Question

Solving a Linear Programming Problem, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. $$ \begin{array}{c}{\text { Objective function: }} \\ {z=5 x+4 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {2 x+2 y \geq 10} \\ {x+2 y \geq 6}\end{array} $$

Step-by-Step Solution

Answer

The optimal points and their corresponding minimum and maximum values of the function \(z = 5x + 4y\), subject to the given constraints, are obtained by subsituting the vertices of the feasible region into the function. However, it requires the sketch of the graph to determine the exact values.

Step 1: Plotting the Constraints

First, we need to plot the inequalities on a graph. Begin with \(x \geq 0\) and \(y \geq 0\), these are simple and denote that we are dealing with the first quadrant of the graph. The third constraint is \(2x + 2y \geq 10\), which simplifies to \(x + y \geq 5\). Similarly, simplify the last constraint as \(x + 2y \geq 6\). Now we plot these lines \(x + y = 5\) and \(x + 2y = 6\) in the graph.

Step 2: Determining the Feasible Region

The region that satisfies all constraints is called the feasible region. It is the intersection of all the half-planes and the quadrants that are defined by each constraint. In this case, the feasible region is the polyhedron bounded by these lines in the first quadrant.

Step 3: Finding the Optimal Points and Values

Plot the objective function \(z = 5x + 4y\). As it's a linear programming problem, the maximum and minimum values of function \(z\) will be at the vertices of the feasible region. By substituting the coordinates of these vertices into function \(z\), we can get the maximum and minimum values.

Key Concepts

Objective Function

The objective function in a linear programming problem is essentially what you want to optimize (maximize or minimize). It comes in the form of a linear equation. In our exercise, the objective function is given by:\[ z = 5x + 4y \]The goal is to find the values of \(x\) and \(y\) that maximize or minimize this equation, subject to the constraints.
The coefficients of \(x\) and \(y\) (which are 5 and 4, respectively) indicate how much each variable influences the function's value. This function represents a line in a 2D space, and our task is to navigate through the constraints to see where this line can achieve its highest or lowest value.
In this method, looking for the highest or lowest point on the plane formed is key. It's like finding the top or bottom point of a mountain, but within the bounds of your constraints.

Constraints

Constraints are conditions that any solution to the linear programming problem must satisfy. They are usually formulated in the form of linear inequalities. In our problem, the restraints guiding our decisions are:

• \(x \geq 0\)
• \(y \geq 0\)
• \(2x + 2y \geq 10\)
• \(x + 2y \geq 6\)

These constraints form boundaries in the graph, limiting where the objective function can achieve its optimal value.
Constraints are critical as they define the feasible region where potential solutions exist. Let's dive deeper into each:

• \(x \geq 0\) and \(y \geq 0\) simply require that solutions are located in the first quadrant where both values are non-negative.
• The inequality \(2x + 2y \geq 10\) simplifies to \(x + y \geq 5\), forming a line where anything above is allowed.
• Similarly, \(x + 2y \geq 6\) sets another line, with viable solutions above this line as well.

Feasible Region

The feasible region is the area on the graph where all the constraints overlap. This is the only area where the solution to the linear programming problem can lie. In the intersection of these constraints, all conditions set by the inequalities meet, creating a polygonal shape in the first quadrant.
Determining the feasible region involves plotting each constraint as a line on the graph and shading the area that satisfies all inequalities. The region is bound by the lines defined by the constraints: \(x + y = 5\) and \(x + 2y = 6\), which both intersect in the first quadrant due to \(x \geq 0\) and \(y \geq 0\).
Once the feasible region is identified, it's crucial to find its vertices, as the minimum and maximum of the objective function will occur at one of these points. In our problem, the resulting shape is a polygon, and by testing the vertices formed by the intersecting lines, you can determine where the objective function reaches its optimal values.