Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 7-8: Sketch the region \(R\) determined by the given constraints, and label its vertices. Describe the set of points for which \(C\) is a maximum on \(R\). $$ \begin{array}{ll} C=6 x+3 y ; & x \geq 2, y \geq 1 \\ 2 x+3 y \leq 19, & x+0.5 y \leq 6.5 \end{array} $$

Step-by-Step Solution

Answer

The maximum value of \( C = 51 \) occurs at the point (8, 1).

Step 1: List the constraints

The region \( R \) is determined by the following constraints: 1. \( x \geq 2 \)2. \( y \geq 1 \)3. \( 2x + 3y \leq 19 \)4. \( x + 0.5y \leq 6.5 \)

Step 2: Sketch the region

Plot each line determined by the equality of the constraints on a coordinate plane:1. \( x = 2 \), a vertical line, for \( x \geq 2 \) shade to the right.2. \( y = 1 \), a horizontal line, for \( y \geq 1 \) shade above.3. \( 2x + 3y = 19 \), rearranged as \( y = \frac{19-2x}{3} \).4. \( x + 0.5y = 6.5 \), rearranged as \( y = 13 - 2x \).Shade the intersection area that satisfies all inequalities, labeling the vertices of this region.

Step 3: Find the vertices of the feasible region

The vertices are the intersections of the lines and borders:1. Intersection of \( x = 2 \) and \( y = 1 \): \( (2, 1) \).2. Intersection of \( x = 2 \) and \( x + 0.5y = 6.5 \): Solve \( 2 + 0.5y = 6.5 \), \( y = 9 \), vertex \( (2, 9) \).3. Intersection of \( 2x + 3y = 19 \) and \( y = 1 \): Solve \( 2x + 3(1) = 19 \), \( 2x = 16 \), \( x = 8 \), vertex \( (8, 1) \).4. Intersection of \( 2x + 3y = 19 \) and \( x + 0.5y = 6.5 \): Solve the system of equations, \( x = 13 - 2x \) and \( 2x + 3y = 19 \). Solve, \( 13 - 2x = y \) and replace in \( 2x + 3(13 - 2x) = 19 \): \( 2x + 39 - 6x = 19 \) \( -4x = -20 \), so \( x = 5 \). Back-substitute to find \( y = 3 \): vertex \( (5, 3) \).

Step 4: Calculate the objective function at vertices

Compute \( C = 6x + 3y \) at each vertex:1. At \( (2, 1) \): \( C = 6(2) + 3(1) = 15 \)2. At \( (2, 9) \): \( C = 6(2) + 3(9) = 39 \)3. At \( (8, 1) \): \( C = 6(8) + 3(1) = 51 \)4. At \( (5, 3) \): \( C = 6(5) + 3(3) = 39 \)

Step 5: Determine maximum value

The maximum value of \( C \) occurs at the vertex with the highest calculated \( C \) value, which is \( C = 51 \) at the point \( (8, 1) \).

Key Concepts

Feasible Region

A feasible region is a set of points that satisfy all the constraints in a linear programming problem. It represents the solution space where all inequalities are met. For our exercise, the feasible region is visualized on a coordinate plane, bordered by the constraints given:

• \( x \geq 2 \)
• \( y \geq 1 \)
• \( 2x + 3y \leq 19 \)
• \( x + 0.5y \leq 6.5 \)

By plotting these constraints as lines and shading the relevant areas, the feasible region is found where all shaded areas overlap, forming a polygon. This area typically looks like a convex shape on the graph. Its vertices are crucial because they are potential "corner points" for finding the optimal solution.

Objective Function

The objective function in linear programming is the formula that needs to be optimized, either maximized or minimized. In this problem, the objective function is given as \( C = 6x + 3y \). Our goal is to find the values of \( x \) and \( y \) within the feasible region that will maximize \( C \).
First, calculate the value of \( C \) at each vertex of the feasible region. These vertices are the intersections of the constraints. This step is crucial as linear programming theories ensure that if an optimal solution exists, it will be found at one of these vertices.

Vertex Optimization

Vertex optimization is the process of evaluating the objective function at each vertex of the feasible region to find the optimal solution. The vertices are determined from the intersections of the constraint lines. In this case, they are:

• \((2, 1)\)
• \((2, 9)\)
• \((8, 1)\)
• \((5, 3)\)

For each vertex, substitute into the objective function \( C = 6x + 3y \) and calculate \( C \). The optimal solution will be the vertex for which \( C \) is the highest. It is important to check each vertex because some vertices may yield better results even if they look less promising at first glance.

Constraints

Constraints are essential elements in linear programming that define the limits within which the solution must exist. They are expressed in the form of inequalities that restrict the values of the variables \( x \) and \( y \).
For the problem at hand, the constraints are:

• \( x \geq 2 \) — implying all solutions must be to the right of the vertical line \( x = 2 \)
• \( y \geq 1 \) — pointing to solutions above the horizontal line \( y = 1 \)
• \( 2x + 3y \leq 19 \) — which must be below the line formed by \( 2x + 3y = 19 \)
• \( x + 0.5y \leq 6.5 \) — limiting solutions under the line \( x + 0.5y = 6.5 \)

These constraints define the boundaries of the feasible region, inside which any potential solution must fall. Understanding constraints is vital to not only solve the problem but also to ensure that the solution makes sense logically and practically within the context of the problem.