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Chapter 5: Problem 23

a. For the given constraints, graph the feasible region and identify the vertices. b. Determine the values of \(x\) and \(y\) that produce the maximum or minimum value of the objective function on the feasible region. c. Determine the maximum or minimum value of the objective function on the feasible region. $$ \begin{array}{l} x \geq 0, y \geq 0 \\ 3 x+y \geq 50 \\ 2 x+y \geq 40 \\ \text { Minimize: } z=3 x+2 y \end{array} $$

Short Answer

Expert verified
Vertices: (0,40), (10,20), (25,0). Minimum value: 70 at (10,20). Maximum value: 80 at (0,40).

Step by step solution

01

- Identify Constraints and Objective Function

The constraints are: 1. \(x \geq 0\)2. \(y \geq 0\)3. \(3x + y \geq 50\)4. \(2x + y \geq 40\)The objective function to minimize is: \(z = 3x + 2y\)
02

- Graph Constraints

Plot the constraints on a graph. Start by plotting the lines: 1. \(3x + y = 50\)2. \(2x + y = 40\)Determine the feasible region, which is where all inequality constraints intersect and satisfy \(x \geq 0\) and \(y \geq 0\).
03

- Determine Vertices of the Feasible Region

Find the intersection points of the constraints to identify the vertices of the feasible region. Solve simultaneously: - Intersection of \(3x + y = 50\) and \(2x + y = 40\): Find \(x = 10\) and \(y = 20\)- Other intersections with axes are not within all constraints, so vertices are: \((0, 40)\), \((10, 20)\), \((25, 0)\)
04

- Evaluate Objective Function at Vertices

Evaluate the objective function \(z = 3x + 2y\) at each vertex: 1. At \((0, 40)\): \(z = 3(0) + 2(40) = 80\)2. At \((10, 20)\): \(z = 3(10) + 2(20) = 70\)3. At \((25, 0)\): \(z = 3(25) + 2(0) = 75\)
05

- Determine Maximum and Minimum Values

From the evaluated values, identify the minimum and maximum: Maximum value: 80 at \((0, 40)\)Minimum value: 70 at \((10, 20)\)

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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 all given constraints. These points form a region on a graph.
This problem has constraints like:
  • \(x \textgreater = 0\)
  • \(y \textgreater = 0\)
  • \(3x + y \textgreater = 50\)
  • \(2x + y \textgreater = 40\)
Combine them by plotting their corresponding lines. The shared region where all these constraints overlap is our feasible region.

Understanding the feasible region is crucial because it contains all possible solutions. Any point outside this region does not satisfy one or more constraints.
Objective Function
The objective function in linear programming is the function you aim to maximize or minimize. It is typically given as a formula, which you need to optimize within the feasible region.
In this exercise, the objective function is:
  • \(z = 3x + 2y\)
Here, we aim to minimize the value of \(z\). To do this, you evaluate the objective function at each vertex of the feasible region.

The point at which the objective function gives the lowest or highest value is the optimal solution to the problem.
Constraints
The constraints limit the feasible region by restricting the values \(x\) and \(y\) can take. These constraints are usually inequalities.
In this problem, we have four constraints:
  • \(x \textgreater = 0\)
  • \(y \textgreater = 0\)
  • \(3x + y \textgreater = 50\)
  • \(2x + y \textgreater = 40\)
Graph each constraint to form lines on a graph. The overlapping area where all inequalities hold true forms the feasible region.

Constraints are essential to consider; they ensure the solutions are valid and real-world applicable.
Vertices
The vertices of the feasible region are the corner points where the boundary lines of the constraints intersect. These vertices are critical since they often contain the optimal solutions.
For this exercise, the vertices are:
  • (0, 40)
  • (10, 20)
  • (25, 0)
These points are determined by solving pairs of linear equations from the constraints. Once identified, evaluate the objective function at these vertices to find the minimum or maximum values.
Graphing
In linear programming, graphing is essential for visualizing constraints and identifying the feasible region. Start by plotting the boundary lines of each constraint.
For instance:
  • \(3x + y = 50\)
  • \(2x + y = 40\)
Next, shade the region that satisfies all inequalities. This shaded area is your feasible region. Finally, mark the vertices of the feasible region. These are the points where the boundary lines intersect within the shaded area.

Graphing helps you visually understand where all constraints meet and where the objective function might reach its optimal value.

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Most popular questions from this chapter

Michelle borrows a total of \(\$ 5000\) in student loans from two lenders. One charges \(4.6 \%\) simple interest and the other charges \(6.2 \%\) simple interest. She is not required to pay off the principal or interest for 3 yr. However, at the end of 3 yr, she will owe a total of \(\$ 762\) for the interest from both loans. How much did she borrow from each lender?

Write a system of inequalities that represents the points inside the triangle with vertices \((-4,-4),(1,1),\) and (5,-1).

Refer to Section 2.5 for a review of linear cost functions and linear revenue functions. A vendor at a carnival sells cotton candy and caramel apples for \(\$ 2.00\) each. The vendor is charged \(\$ 100\) to set up his booth. Furthermore, the vendor's average cost for each product he produces is approximately \(\$ 0.75\) a. Write a linear cost function representing the cost \(C(x)\) (in \(\$$ ) to produce \)x\( products. b. Write a linear revenue function representing the revenue \)R(x)\( (in \$) for selling \)x$ products. c. Determine the number of products to be produced and sold for the vendor to break even. d. If 60 products are sold, will the vendor make money or lose money?

A couple has \(\$ 60,000\) to invest for retirement. They plan to put \(x\) dollars in stocks and \(y\) dollars in bonds. For parts (a)-(d), write an inequality to represent the given statement. a. The total amount invested is at most \(\$ 60,000\). b. The couple considers stocks a riskier investment, so they want to invest at least twice as much in bonds as in stocks. c. The amount invested in stocks cannot be negative. d. The amount invested in bonds cannot be negative. e. Graph the solution set to the system of inequalities from parts (a)-(d).

Monique and Tara each make an ice-cream sundae. Monique gets 2 scoops of Cherry ice-cream and 1 scoop of Mint Chocolate Chunk ice-cream for a total of \(43 \mathrm{~g}\) of fat. Tara has 1 scoop of Cherry and 2 scoops of Mint Chocolate Chunk for a total of \(47 \mathrm{~g}\) of fat. How many grams of fat does 1 scoop of each type of ice cream have?
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