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Chapter 12: Problem 9

Solve each linear programming problem. Maximize \(z=2 x+y\) subject to the constraints \(x \geq 0, \quad y \geq 0, \quad x+y \leq 6, \quad x+y \geq 1\)

Short Answer

Expert verified
The maximum value of z is 11 at the point (5, 1).

Step by step solution

01

- Identify the Objective Function

The objective function we need to maximize is given by: z = 2x + y
02

- Identify the Constraints

The problem is subject to the following constraints: \( x \geq 0 \), \( y \geq 0 \), \(x+y \leq 6 \), and \( x+y \geq 1 \)
03

- Graph the Inequalities

Sketch the constraints on a graph: - The line \( x + y = 6 \) divides the plane into two regions. The shaded region will be below this line. - The line \( x + y = 1 \) divides the plane below. The shaded region will be above this line. - The non-negative constraints \( x \geq 0 \) and \( y \geq 0 \) help to keep the solution in the first quadrant.
04

- Find the Feasible Region

The feasible region is the intersection of all the shaded regions defined by the constraints. This region is bounded by the lines \( x + y = 6 \), \( x + y = 1 \), and the \( x \) and \( y\)-axes.
05

- Identify Corner Points

Identify the corner points (vertices) of the feasible region. These points are the intersections of the lines: - (0, 1) where \( x + y = 1 \) and \( x = 0 \) - (0, 6) where \( x + y = 6 \) and \( x = 0 \) - (1, 0) where \( x + y = 1 \) and \( y = 0 \) - (5, 1) where \( x + y = 6 \) and \( x + y = 1 \)
06

- Evaluate the Objective Function at Each Corner Point

Substitute each vertex into the objective function to find the value of \( z = 2x + y \): - At (0, 1): \( z = 2(0) + 1 = 1 \) - At (0, 6): \( z = 2(0) + 6 = 6 \) - At (1, 0): \( z = 2(1) + 0 = 2 \) - At (5, 1): \( z = 2(5) + 1 = 11 \)
07

- Determine the Maximum Value

The maximum value of \( z \) is the highest value obtained in the evaluations. This occurs at point (5, 1) where \( z = 11 \).

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Key Concepts

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

Objective Function
The objective function is a mathematical expression that defines what you want to maximize or minimize in a linear programming problem. In this exercise, the objective function is given by:
\(z = 2x + y\).
Here, the goal is to find the values of \(x\) and \(y\) that will maximize the value of \(z\). Essentially, the objective function serves as your 'target' that you aim to optimize subject to certain limitations called constraints.
Constraints
Constraints are conditions that must be met for the solution to be valid. In our exercise, the constraints are:
  • \(x \geq 0\)
  • \(y \geq 0\)
  • \(x + y \leq 6\)
  • \(x + y \geq 1\)
These constraints limit the values that \(x\) and \(y\) can take.
Visualizing these in a graph helps to see where these constraints overlap. This overlap area is crucial as it contains all possible solutions that meet all the constraints.
Feasible Region
The feasible region is the area on the graph where all the constraints overlap. It represents all possible values of \(x\) and \(y\) that satisfy all the given constraints.
To determine the feasible region for our exercise, we plot the constraints on a graph:
  • The line \(x + y = 6\) and the area below it.
  • The line \(x + y = 1\) and the area above it.
  • The non-negative constraints \(x \geq 0\) and \(y \geq 0\), restricting the solution to the first quadrant.
The intersection of these regions gives us the feasible region, which is bounded by the lines \(x + y = 6\), \(x + y = 1\), and the \(x\) and \(y\)-axes.
Corner Points
Corner points, or vertices, are the points where the boundaries of the feasible region intersect. These points are vital to solving a linear programming problem because the maximum or minimum value of the objective function occurs at one of these points.
For this exercise, the corner points are:
  • (0, 1) where \(x + y = 1\) and \(x = 0\)
  • (0, 6) where \(x + y = 6\) and \(x = 0\)
  • (1, 0) where \(x + y = 1\) and \(y = 0\)
  • (5, 1) where \(x + y = 6\) and \(x + y = 1\)
You then evaluate the objective function at each of these points to find which one gives the maximum value. In our case, substituting these points into \(z = 2x + y\) shows that the maximum value is 11 at point (5, 1).

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Most popular questions from this chapter

A dietitian at Palos Community Hospital wants a patient to have a meal that has 78 grams \((\mathrm{g})\) of protein, \(59 \mathrm{~g}\) of carbohydrates, and 75 milligrams (mg) of vitamin A. The hospital food service tells the dietitian that the dinner for today is salmon steak, baked eggs, and acorn squash. Each serving of salmon steak has \(30 \mathrm{~g}\) of protein, \(20 \mathrm{~g}\) of carbohydrates, and \(2 \mathrm{mg}\) of vitamin \(\mathrm{A}\). Each serving of baked eggs contains \(15 \mathrm{~g}\) of protein, \(2 \mathrm{~g}\) of carbohydrates, and \(20 \mathrm{mg}\) of vitamin A. Each serving of acorn squash contains \(3 \mathrm{~g}\) of protein, \(25 \mathrm{~g}\) of carbohydrates, and \(32 \mathrm{mg}\) of vitamin \(\mathrm{A} .\) How many servings of each food should the dietitian provide for the patient?

A banquet hall offers two types of tables for rent: 6 -person rectangular tables at a cost of \(\$ 28\) each and 10 -person round tables at a cost of \(\$ 52\) each. Kathleen would like to rent the hall for a wedding banquet and needs tables for 250 people. The hall can have a maximum of 35 tables, and the hall has only 15 rectangular tables available. How many of each type of table should be rented to minimize cost and what is the minimum cost?

Use the following matrices. Determine whether the given expression is defined. If it is defined, express the result as a single matrix; if it is not, write "not defined" $$ A=\left[\begin{array}{rrr} 0 & 3 & -5 \\ 1 & 2 & 6 \end{array}\right] \quad B=\left[\begin{array}{rrr} 4 & 1 & 0 \\ -2 & 3 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & 1 \\ 6 & 2 \\ -2 & 3 \end{array}\right] $$ CB

Solve each system. Use any method you wish. $$ \left\\{\begin{array}{r} 2 x^{2}-x y+y^{2}=8 \\ x y=4 \end{array}\right. $$

Nutrition A dietitian wishes a patient to have a meal that has 66 grams (g) of protein, 94.5 g of carbohydrates, and 910 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is chicken, corn, and \(2 \%\) milk. Each serving of chicken has \(30 \mathrm{~g}\) of protein, \(35 \mathrm{~g}\) of carbohydrates, and \(200 \mathrm{mg}\) of calcium. Each serving of corn has \(3 \mathrm{~g}\) of protein, \(16 \mathrm{~g}\) of carbohydrates, and \(10 \mathrm{mg}\) of calcium. Each glass of \(2 \%\) milk has \(9 \mathrm{~g}\) of protein, \(13 \mathrm{~g}\) of carbohydrates, and \(300 \mathrm{mg}\) of calcium. How many servings of each food should the dietitian provide for the patient?
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