28. Minimize and maximize z = 400x + 100y %3D subject to 3x + y 24 x +y 16 x + 3y 2 30 x, y > 0
28. Minimize and maximize z = 400x + 100y %3D subject to 3x + y 24 x +y 16 x + 3y 2 30 x, y > 0
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
![### Linear Programming Problem
**Objective:**
Minimize and maximize the function:
\[ z = 400x + 100y \]
**Subject to the constraints:**
1. \( 3x + y \geq 24 \)
2. \( x + y \geq 16 \)
3. \( x + 3y \geq 30 \)
4. \( x, y \geq 0 \)
### Graphical Representation:
The graph provides a coordinate plane with a grid where you can plot the feasible region determined by the constraints. Each inequality will translate into a line on the graph, dividing the plane into regions of allowed and disallowed values for \( x \) and \( y \). The solution to the problem will be found at the vertices of the feasible region, which is the intersection of the constraints plotted on this grid.
- **Axes:** The horizontal axis represents variable \( x \) and the vertical axis represents \( y \).
- **Grid:** Each square on the grid can be used to estimate the values of \( x \) and \( y \).
This exercise involves using graphical methods to find where the constraints intersect, forming a polygonal feasible region. The optimal solutions (both minimum and maximum) are found by evaluating the objective function \( z \) at these points of intersection.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F703a50ff-8f5d-40ba-8eea-f13205fb216a%2Ffd09f3c9-cabc-4ee8-a48c-794a27497b9c%2Fxrvvoge.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Programming Problem
**Objective:**
Minimize and maximize the function:
\[ z = 400x + 100y \]
**Subject to the constraints:**
1. \( 3x + y \geq 24 \)
2. \( x + y \geq 16 \)
3. \( x + 3y \geq 30 \)
4. \( x, y \geq 0 \)
### Graphical Representation:
The graph provides a coordinate plane with a grid where you can plot the feasible region determined by the constraints. Each inequality will translate into a line on the graph, dividing the plane into regions of allowed and disallowed values for \( x \) and \( y \). The solution to the problem will be found at the vertices of the feasible region, which is the intersection of the constraints plotted on this grid.
- **Axes:** The horizontal axis represents variable \( x \) and the vertical axis represents \( y \).
- **Grid:** Each square on the grid can be used to estimate the values of \( x \) and \( y \).
This exercise involves using graphical methods to find where the constraints intersect, forming a polygonal feasible region. The optimal solutions (both minimum and maximum) are found by evaluating the objective function \( z \) at these points of intersection.
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