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Chapter 7: Problem 32

In Exercises 29-34, 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. Objective function: \( z = x + y \) Constraints: \( \hspace{1cm} x \ge 0 \) \( \hspace{1cm} y \ge 0 \) \( -x + y \le 0 \) \( -3x + y \ge 3 \)

Short Answer

Expert verified
The feasible region forms a triangle in the first quadrant with vertices at (0,0), (0,3) and (1,1). The minimum value of the objective function is 0, occurring at (0,0), and the maximum value is 3, occurring at (0,3).

Step by step solution

01

Sketch the feasible region

Use the given constraints to sketch the feasible region, which is a 2-dimensional region on the coordinate plane. Note that the constraint \(x \ge 0\) implies that all feasible points lie to the right of the y-axis. Similarly, \(y \ge 0\) means points are above the x-axis. The constraints, \( -x + y \le 0 \) and \( -3x + y \ge 3 \), represent the lines \( y = x \) and \( y = 3x + 3 \) respectively.
02

Find the intersection points

The two lines intersect the axes at (0,0), (0,3) and (1,1), forming a triangle in the first quadrant. Therefore, the feasible region under the constraints would be the triangle formed by these intersection points.
03

Find the minimum and maximum values

Substitute the coordinates of the vertices of the feasible region into the objective function \(z = x + y\) to find the maximum and minimum value. For (0,0): \(z=0\). For (0,3): \(z=3\). For (1,1): \(z=2\). Hence the minimum value of z is \(0\) at \( (0,0) \) and maximum is \(3\) at \( (0,3) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Feasible Region
In linear programming, the feasible region is the set of all possible points that satisfy the problem's constraints and is where the solution to the linear programming problem must lie. It's the area where all the conditions of a given problem simultaneously apply.

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Most popular questions from this chapter

In Exercises 55-60, use a graphing utility to graph the solution set of the system of inequalities. \( \left\\{\begin{array}{l} y < -x^2 + 2x + 3\\\ y > x^2 - 4x + 3\end{array}\right. \)

A dietitian is asked to design a special dietary supplement using two different foods. Each ounce of food \( X \) contains 20 units of calcium, 15 units of iron, and 10 units of vitamin \( B \). Each ounce of food \( Y \) contains 10 units of calcium, 10 units of iron, and 20 units of vitamin \( B \). The minimum daily requirements of the diet are 300 units of calcium, 150 units of iron, and 200 units of vitamin \( B \). (a) Write a system of inequalities describing the different amounts of food \( X \) and food \( Y \) that can be used. (b) Sketch a graph of the region corresponding to the system in part (a). (c) Find two solutions of the system and interpret their meanings in the context of the problem.

In Exercises 7-20, sketch the graph of the inequality. \( 5x + 3y \ge -15 \)

In Exercises 29-34, 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. Objective function: \( z = -x + 2y \) Constraints: \( \hspace{1cm} x \ge 0 \) \( \hspace{1cm} y \ge 0 \) \( \hspace{1cm} x \le 10 \) \( x + y \le 7 \)

In Exercises 41-54, sketch the graph and label the vertices of the solution set of the system of inequalities. \( \left\\{\begin{array}{l} 4x^2 + y \ge 2\\\ \hspace{1cm} x \le 1\\\ \hspace{1cm} y \le 1\end{array}\right. \)
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