Problem 13 - College Algebra

Question

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$\begin{aligned}&z=10 x+12 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\x+y \leq 7 \\\2 x+y \leq 10 \\\2 x+3 y \leq 18\end{array}\right.\end{aligned}$$

Step-by-Step Solution

Answer

The maximum value of the objective function is 78, at the point (3, 4)

Step 1: Graphing the System of Inequalities

First, graph all the given inequalities:(a) \(x \geq 0\) and \(y \geq 0\) represent the first quadrant which includes the origin. (b) For \(x + y \leq 7\), y-intercept is at 7 (when x=0, y=7) and x-intercept is at 7 (when y=0, x=7).(c) For \(2x + y \leq 10\), y-intercept is at 10 (when x=0, y=10) and x-intercept is at 5 (when y=0, x=5).(d) For \(2x + 3y \leq 18\), y-intercept is at 6 (when x=0, y=6) and x-intercept is at 9 (when y=0, x=9).The shaded region represents the common area for all the inequalities.

Step 2: Find the value of the Objective Function at each corner of the Region

The vertices of the feasible region as determined from the graph are (0,0), (0,6), (2,5) and (3,4). We substitute these points into the objective function \(z=10x+12y\).(a) For (0,0), the value of \(z\) = 10*0 + 12*0 = 0.(b) For (0,6), the value of \(z\) = 10*0 + 12*6 =72.(c) For (2,5), the value of \(z\) = 10*2 + 12*5 =76.(d) For (3,4), the value of \(z\) = 10*3 + 12*4 =78.

Step 3: Determine the Maximum Value of the Objective Function and the Corresponding values of x and y

The maximum value of the objective function \(z\), by comparing results from Step 2, is 78, which is achieved when \(x=3\) and \(y=4\). Hence, values of \(x\) and \(y\) for which the maximum occurs are \(x=3\) and \(y=4\).

Key Concepts

Graphing Inequalities

Understanding how to graph inequalities is crucial in linear programming. It's similar to plotting equations, but instead of equal signs, you've got inequalities symbolizing a range of possible solutions. Let's break down the core steps:

• Start by identifying the inequality's equation as if it were an equality, such as turning \(x + y \leq 7\) into \(x + y = 7\).
• Graph this line on a coordinate plane.
• Next, determine which side of the line fits the inequality. You can test a point not on the line (like \(0,0\)) to see if it satisfies the inequality. If it does, shade that side of the line.
• Repeat this for each inequality. The overlapping shaded region is of special interest.

This region will show us where the solutions to all inequalities intersect, indicating possible solutions for the system.

Objective Function Optimization

To optimize the objective function in a linear programming problem means to find its maximum or minimum value under the given constraints. In our exercise, the objective function is \(z=10x+12y\), where we aim to maximize \(z\).

After graphing the inequalities and identifying the feasible region, the next step is to evaluate the objective function at each vertex (or 'corner') of this region. These corners are typically where the maximum or minimum values occur, as per the theory of linear programming. By comparing the value of \(z\) at these points, we can pinpoint where the function is optimized.

Feasible Region Determination

The feasible region is the set of all possible points that satisfy all the problem's constraints. It's determined by graphing the system of inequalities. Here's what to keep in mind:

• The feasible region should always lie within the boundaries set by the inequalities.
• If an inequality includes an equals sign (\(\leq\) or \((\geq\)), then the line itself is part of the feasible region.
• The region is often a polygon-shaped area on the graph. If it's bounded (closed and finite), as in our textbook example, then the problem has a solution within that area.

Identifying this region correctly is key because it contains the answer to what the question is asking: the optimal values for \(x\) and \(y\) that will maximize or minimize our objective function.

System of Linear Inequalities

A system of linear inequalities consists of several linear inequalities that together define the constraints of a linear programming problem. When dealing with multiple inequalities:

• Each inequality divides the plane into two halves. We need the half that satisfies the inequality.
• Where these halves intersect, that's our feasible region.
• The system is considered in its entirety, meaning we look for a solution that satisfies all inequalities simultaneously.

It's like piecing together a puzzle where each piece's placement is essential, and when you put them all together, they reveal the 'big picture'—in this case, the feasible region where our optimal values for \(x\) and \(y\) lie.