Solve the linear programming problem by sketching the region and labeling the vertices, deciding whether a solution exists, and then finding it if it does exist. (If an answer does not exist, enter DNE.) Minimize C = 10x + 40y S2x + 5y z 20 \x 2 0, y 2 0 Subject to
Solve the linear programming problem by sketching the region and labeling the vertices, deciding whether a solution exists, and then finding it if it does exist. (If an answer does not exist, enter DNE.) Minimize C = 10x + 40y S2x + 5y z 20 \x 2 0, y 2 0 Subject to
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
9h
![**Linear Programming Problem**
**Objective:**
Solve the linear programming problem by sketching the region, labeling the vertices, deciding whether a solution exists, and then finding it if it does exist. (If an answer does not exist, enter DNE.)
**Problem:**
Minimize: \( C = 10x + 40y \)
Subject to:
\[
\begin{align*}
2x + 5y & \geq 20 \\
x & \geq 0 \\
y & \geq 0 \\
\end{align*}
\]
**Instructions:**
1. **Graph the Constraints**: Sketch the feasible region by plotting the lines represented by the inequalities. The feasible region is the area that satisfies all the given constraints.
2. **Find Vertices**: Identify and label the vertices (corner points) of the feasible region. These are points where the boundary lines intersect and could potentially minimize \( C \).
3. **Check for Solution**: Examine the feasible region to determine if it is bounded and if an optimal solution exists.
4. **Calculate Minimum Value**: If an optimal solution exists, substitute the vertices into the objective function \( C = 10x + 40y \) to find the minimum value.
5. **Determine Result**: If a solution exists, record the minimum value of \( C \). If no solution exists, input DNE (Does Not Exist).
Use this framework to approach linear programming problems systematically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d8e838a-6c8b-4033-a5e3-a56465c5aa54%2F775a09ab-aa0a-471f-874e-5e50775ab930%2Fjop2y_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Programming Problem**
**Objective:**
Solve the linear programming problem by sketching the region, labeling the vertices, deciding whether a solution exists, and then finding it if it does exist. (If an answer does not exist, enter DNE.)
**Problem:**
Minimize: \( C = 10x + 40y \)
Subject to:
\[
\begin{align*}
2x + 5y & \geq 20 \\
x & \geq 0 \\
y & \geq 0 \\
\end{align*}
\]
**Instructions:**
1. **Graph the Constraints**: Sketch the feasible region by plotting the lines represented by the inequalities. The feasible region is the area that satisfies all the given constraints.
2. **Find Vertices**: Identify and label the vertices (corner points) of the feasible region. These are points where the boundary lines intersect and could potentially minimize \( C \).
3. **Check for Solution**: Examine the feasible region to determine if it is bounded and if an optimal solution exists.
4. **Calculate Minimum Value**: If an optimal solution exists, substitute the vertices into the objective function \( C = 10x + 40y \) to find the minimum value.
5. **Determine Result**: If a solution exists, record the minimum value of \( C \). If no solution exists, input DNE (Does Not Exist).
Use this framework to approach linear programming problems systematically.
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