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Chapter 3: Problem 25

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{r} x+y \leq 6 \\ 0 \leq x \leq 3 \\ y \geq 0 \end{array} $$

Short Answer

Expert verified
The solution set for the given system of inequalities is a bounded trapezoidal region with vertices at points (0, 0), (3, 0), (3, 3), and (0, 6).

Step by step solution

01

Graph the inequalities

First, we need to graph each inequality on the coordinate plane. We can rewrite each inequality as an equation to help with graphing: 1. \(x + y = 6\) - The boundary is a straight line that passes through the points (0, 6) and (6, 0). 2. \(x = 0\) - The boundary is a vertical line along the y-axis. 3. \(x = 3\) - The boundary is a vertical line passing through the point (3, 0). 4. \(y = 0\) - The boundary is a horizontal line along the x-axis.
02

Shade the feasible regions

Now that we have graphed the boundaries of the inequalities, we need to shade the feasible regions that satisfy each inequality: 1. \(x + y \leq 6\) - Shade the region below the line. 2. \(0 \leq x \leq 3\) - Shade the region between the vertical lines at x = 0 and x = 3. 3. \(y \geq 0\) - Shade the region above the line y = 0.
03

Determine the intersection of the feasible regions

The solution set of the system of inequalities is the intersection of the feasible regions from each inequality. In the coordinate plane, this intersection is a trapezoidal region with vertices at points (0, 0), (3, 0), (3, 3), and (0, 6).
04

Check if the solution set is bounded or unbounded

As the intersection of the feasible regions is a closed region without the possibility of extending infinitely in any direction, the solution set is bounded. So, the solution set for the given system of inequalities is a bounded trapezoidal region with vertices at points (0, 0), (3, 0), (3, 3), and (0, 6).

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Most popular questions from this chapter

Manufacturing has a division that produces two models of fireplace grates, model A and model B. To produce each model \(\mathrm{A}\) grate requires \(3 \mathrm{lb}\) of cast iron and \(6 \mathrm{~min}\) of labor. To produce each model B grate requires \(4 \mathrm{lb}\) of cast iron and 3 min of labor. The profit for each model A grate is \(\$ 2.00\), and the profit for each model B grate is \(\$ 1.50\). If \(1000 \mathrm{lb}\) of cast iron and 20 labor-hours are available for the production of fireplace grates per day, how many grates of each model should the division produce in order to maximize Kane's profit? What is the optimal profit?

Consider the linear programming problem $$ \begin{array}{rr} \text { Maximize } & P=2 x+7 y \\ \text { subject to } & 2 x+y \geq 8 \\ & x+y \geq 6 \\ x & \geq 0, y \geq 0 \end{array} $$ a. Sketch the feasible set \(S\). b. Find the corner points of \(S\). c. Find the values of \(P\) at the corner points of \(S\) found in part (b). d. Show that the linear programming problem has no (optimal) solution. Does this contradict Theorem 1 ?

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