Problem 30
Question
Use a graphing utility to sketch the region determined by the constraints. Then determine the maximum value of the objective function subject to the contraints. Objective Function \(\quad z=30 x+20 y\) $$\begin{array}{ll} \text { Constraints } & x \geq 0, y \geq 0 \\ & 2 x+y \leq 14 \\ & 3 x+y \leq 18 \end{array}$$
Step-by-Step Solution
Verified Answer
The maximum value of the objective function subject to the given constraints is 300, occurring at point (4,10).
1Step 1: Sketch the feasible region
First, graph the inequalities \(2x + y \leq 14\), \(3x + y \leq 18\), \(x \geq 0\), and \(y \geq 0\). The feasible region is where all these inequalities intersect.
2Step 2: Determine the vertices of the feasible region
After sketching the feasible region, identify its vertices. The graphical intersection points of the constraints are at (0,0), (0,14), (4,10), and (6,0).
3Step 3: Calculate objective function values
Substitute the vertex coordinates into the objective function \(z = 30x + 20y\). The corresponding Z-values are: Z(0,0) = 0, Z(0,14) = 280, Z(4,10) = 300, and Z(6,0) = 180.
4Step 4: Pick the maximum value
From the calculated Z-values, the maximum value is 300. Thus the maximum value occurs at point (4,10).
Key Concepts
Objective FunctionConstraintsFeasible RegionGraphing Utility
Objective Function
In linear programming, the objective function is the formula you want to optimize. This can either be maximizing or minimizing a particular value. In our exercise, the objective function is given as \( z = 30x + 20y \). Here, \( z \) represents the value you are trying to maximize (or minimize), while \( x \) and \( y \) are variables influencing the outcome.
The coefficients (30 and 20 in this case) indicate how much each variable contributes to the overall value of \( z \). When solving such problems, it's important to remember that the goal is to find the combination of \( x \) and \( y \) that produces the maximum possible value for \( z \), while also considering any restrictions placed on \( x \) and \( y \).
The coefficients (30 and 20 in this case) indicate how much each variable contributes to the overall value of \( z \). When solving such problems, it's important to remember that the goal is to find the combination of \( x \) and \( y \) that produces the maximum possible value for \( z \), while also considering any restrictions placed on \( x \) and \( y \).
- Objective function: formula to optimize
- Maximize or minimize its value
- Coefficients show impact on the result
Constraints
Constraints are conditions that limit the values that the decision variables \( x \) and \( y \) can take on. Constraints ensure that the solution is realistic and necessarily feasible under certain conditions. In our exercise, the given constraints are:
- \( x \geq 0 \)
- \( y \geq 0 \)
- \( 2x + y \leq 14 \)
- \( 3x + y \leq 18 \)
These conditions restrict values so that all solutions must not only be greater than or equal to zero but must also fall below certain linear boundaries formed by the inequalities. By visualizing these constraints, each inequality represents a half-plane on a graph and the solution set—the feasible region—is where all individual conditions overlap. This is a key step to narrowing down the graph to realistic and practical solutions.
- \( x \geq 0 \)
- \( y \geq 0 \)
- \( 2x + y \leq 14 \)
- \( 3x + y \leq 18 \)
These conditions restrict values so that all solutions must not only be greater than or equal to zero but must also fall below certain linear boundaries formed by the inequalities. By visualizing these constraints, each inequality represents a half-plane on a graph and the solution set—the feasible region—is where all individual conditions overlap. This is a key step to narrowing down the graph to realistic and practical solutions.
- Define boundaries for variables
- Ensure solutions are realistic
- Graphical representation needed for overlap
Feasible Region
The feasible region is the area on a graph where all constraints overlap. It represents all possible combinations of \( x \) and \( y \) that satisfy every constraint in a system. In simpler terms, it's like finding all the spots where every imagined condition on \( x \) and \( y \) come together to form a workable solution area.
To find this region, you first graph the constraints. Each inequality divides the plane into two halves. The feasible region is where the solution to each inequality exists simultaneously. In the given problem, this means shading the area that fits within the equations \( 2x + y \leq 14 \) and \( 3x + y \leq 18 \), while also being above \( x \geq 0 \) and \( y \geq 0 \). The vertices of this region play a crucial role because at least one vertex will provide the maximum or minimum value of the objective function.
To find this region, you first graph the constraints. Each inequality divides the plane into two halves. The feasible region is where the solution to each inequality exists simultaneously. In the given problem, this means shading the area that fits within the equations \( 2x + y \leq 14 \) and \( 3x + y \leq 18 \), while also being above \( x \geq 0 \) and \( y \geq 0 \). The vertices of this region play a crucial role because at least one vertex will provide the maximum or minimum value of the objective function.
- Intersection of all constraints
- Represents possible solutions
- Key focus area for solution
Graphing Utility
A graphing utility is a technological tool that helps depict mathematical functions and inequalities visually. For students struggling with manual graphing or for those needing a more precise and error-free approach, a graphing utility comes in handy. This utility can quickly convert equations into a visual graph and reveals the feasible region by highlighting where all constraints overlap.
When using a graphing utility, input your constraints, and allow the program to sketch the feasible area. It helps identify and calculate critical points, such as the vertices, where potential solutions to the optimization problem exist. These technological tools not only ease the process of plotting multiple functions but also enhance understanding by providing immediate visual feedback that complements theoretical learning.
When using a graphing utility, input your constraints, and allow the program to sketch the feasible area. It helps identify and calculate critical points, such as the vertices, where potential solutions to the optimization problem exist. These technological tools not only ease the process of plotting multiple functions but also enhance understanding by providing immediate visual feedback that complements theoretical learning.
- Depicts functions visually
- Efficiently finds intersections
- Enhances understanding with visual feedback
Other exercises in this chapter
Problem 30
In Exercises \(19-30,\) solve each system by the addition method. \(5 x=6 y+40\) \(2 y=8-3 x\)
View solution Problem 30
Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&x+y \leq 4\\\&y \geq 2 x-4\end{aligned} $$
View solution Problem 30
Solve each system by the method of your choice. $$\begin{aligned} &x+y^{2}=4\\\ &x^{2}+y^{2}=16 \end{aligned}$$
View solution Problem 31
In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to
View solution