Find the maximum and minimum values of the function and the values of x and y where they occur. F = 5x +36y, subject to 6x + 4y = 35, 4x + 6y≤30, x≥0, y 20. What is the maximum value of the funct (Simplify your answer.)
Find the maximum and minimum values of the function and the values of x and y where they occur. F = 5x +36y, subject to 6x + 4y = 35, 4x + 6y≤30, x≥0, y 20. What is the maximum value of the funct (Simplify your answer.)
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
Find the max
Find the min
And find the values of x and y where they occur.
Four part question
![### Linear Programming Problem
**Problem Statement:**
Find the maximum and minimum values of the function \(F\) and the values of \(x\) and \(y\) where they occur.
**Function to Optimize:**
\[ F = 5x + 36y \]
**Subject to Constraints:**
\[ 6x + 4y \leq 35 \]
\[ 4x + 6y \leq 30 \]
\[ x \geq 0 \]
\[ y \geq 0 \]
**Question:**
What is the maximum value of the function?
**Instructions:**
Simplify your answer.
---
#### Explanation of Constraints:
1. The first constraint, \(6x + 4y \leq 35\), represents a linear inequality which might describe a resource limitation or boundary in a feasible region.
2. The second constraint, \(4x + 6y \leq 30\), represents another linear inequality which works similarly.
3. \(x \geq 0\) and \(y \geq 0\) ensure that \(x\) and \(y\) are non-negative, meaning they must fall within the first quadrant of the Cartesian coordinate system.
For educational purposes, it is essential to graph the inequalities to find the feasible region, where all constraints are satisfied simultaneously. The vertices (corner points) of the feasible region should then be determined, as these typically provide the values where the function \(F\) will achieve its maximum and minimum, assuming such points exist within the feasible region.
### Graphical Representation:
To solve the problem graphically, follow these steps:
1. **Graph the Inequalities:**
- Translate each inequality into a straight line.
- Determine the shading direction (typically below the line for \(\leq\) and above for \(\geq\)).
2. **Identify the Feasible Region:**
- The feasible region is where all shaded areas from the inequalities overlap and it lies in the first quadrant.
3. **Find the Vertices:**
- Calculate the intersection points of the boundary lines and the axes.
- Check where the lines intersect by solving the system of equations formed by each pair of lines.
4. **Evaluate the Function \(F = 5x + 36y\):**
- Evaluate \(F\) at each vertex of the feasible region to find the maximum and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f2946d2-02a8-4a61-8c67-ece11f9611dd%2F5669c5f8-6ab0-47ca-9e8a-3c21594199f3%2Fay2ytm8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Programming Problem
**Problem Statement:**
Find the maximum and minimum values of the function \(F\) and the values of \(x\) and \(y\) where they occur.
**Function to Optimize:**
\[ F = 5x + 36y \]
**Subject to Constraints:**
\[ 6x + 4y \leq 35 \]
\[ 4x + 6y \leq 30 \]
\[ x \geq 0 \]
\[ y \geq 0 \]
**Question:**
What is the maximum value of the function?
**Instructions:**
Simplify your answer.
---
#### Explanation of Constraints:
1. The first constraint, \(6x + 4y \leq 35\), represents a linear inequality which might describe a resource limitation or boundary in a feasible region.
2. The second constraint, \(4x + 6y \leq 30\), represents another linear inequality which works similarly.
3. \(x \geq 0\) and \(y \geq 0\) ensure that \(x\) and \(y\) are non-negative, meaning they must fall within the first quadrant of the Cartesian coordinate system.
For educational purposes, it is essential to graph the inequalities to find the feasible region, where all constraints are satisfied simultaneously. The vertices (corner points) of the feasible region should then be determined, as these typically provide the values where the function \(F\) will achieve its maximum and minimum, assuming such points exist within the feasible region.
### Graphical Representation:
To solve the problem graphically, follow these steps:
1. **Graph the Inequalities:**
- Translate each inequality into a straight line.
- Determine the shading direction (typically below the line for \(\leq\) and above for \(\geq\)).
2. **Identify the Feasible Region:**
- The feasible region is where all shaded areas from the inequalities overlap and it lies in the first quadrant.
3. **Find the Vertices:**
- Calculate the intersection points of the boundary lines and the axes.
- Check where the lines intersect by solving the system of equations formed by each pair of lines.
4. **Evaluate the Function \(F = 5x + 36y\):**
- Evaluate \(F\) at each vertex of the feasible region to find the maximum and
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