Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To maximize the objective function [tex]\( P = 3x + 4y \)[/tex] given the constraints:
1. [tex]\( x \geq 0 \)[/tex]
2. [tex]\( y \geq 0 \)[/tex]
3. [tex]\( 2x + 2y \geq 4 \)[/tex]
4. [tex]\( x + y \leq 8 \)[/tex]
we will follow these steps:
### Step 1: Rewrite the Constraints for Simplicity
- [tex]\( 2x + 2y \geq 4 \)[/tex] can be simplified to [tex]\( x + y \geq 2 \)[/tex]
- So, the constraints now are:
1. [tex]\( x \geq 0 \)[/tex]
2. [tex]\( y \geq 0 \)[/tex]
3. [tex]\( x + y \geq 2 \)[/tex]
4. [tex]\( x + y \leq 8 \)[/tex]
### Step 2: Identify the Feasible Region
We plot the constraints on the xy-plane to find the feasible region where all constraints are satisfied.
1. [tex]\( x \geq 0 \)[/tex]: This means only consider the right half of the y-axis (including the y-axis itself).
2. [tex]\( y \geq 0 \)[/tex]: This means only consider the upper half of the x-axis (including the x-axis itself).
3. [tex]\( x + y \geq 2 \)[/tex]: This is a line that extends from (2, 0) to (0, 2) and continues infinitely up to the right. The feasible region will be above this line.
4. [tex]\( x + y \leq 8 \)[/tex]: This is a line that extends from (8, 0) to (0, 8) and the feasible region will be below this line.
The feasible region is thus bounded by:
1. The line [tex]\( x + y = 2 \)[/tex]
2. The line [tex]\( x + y = 8 \)[/tex]
3. The x-axis
4. The y-axis
### Step 3: Determine the Corner Points
Find the corner points of the feasible region by determining the intersection points of the lines and axes:
1. Intersection of [tex]\( x + y = 2 \)[/tex] and [tex]\( x = 0 \)[/tex]: [tex]\( (0, 2) \)[/tex]
2. Intersection of [tex]\( x + y = 2 \)[/tex] and [tex]\( y = 0 \)[/tex]: [tex]\( (2, 0) \)[/tex]
3. Intersection of [tex]\( x + y = 8 \)[/tex] and [tex]\( x = 0 \)[/tex]: [tex]\( (0, 8) \)[/tex]
4. Intersection of [tex]\( x + y = 8 \)[/tex] and [tex]\( y = 0 \)[/tex]: [tex]\( (8, 0) \)[/tex]
5. Intersection of [tex]\( x + y = 2 \)[/tex] and [tex]\( x + y = 8 \)[/tex]: Solve the equations simultaneously:
[tex]\[ x + y = 2 \\ x + y = 8 \][/tex]
These two lines are parallel and never intersect.
The feasible region is a quadrilateral with potential corner points: [tex]\( (0, 2) \)[/tex], [tex]\( (2, 0) \)[/tex], [tex]\( (8, 0) \)[/tex], and [tex]\( (0, 8) \)[/tex].
### Step 4: Evaluate the Objective Function at Each Corner Point
Calculate the value of [tex]\( P = 3x + 4y \)[/tex] at each corner point:
1. At [tex]\( (0, 2) \)[/tex]:
[tex]\[ P = 3(0) + 4(2) = 0 + 8 = 8 \][/tex]
2. At [tex]\( (2, 0) \)[/tex]:
[tex]\[ P = 3(2) + 4(0) = 6 + 0 = 6 \][/tex]
3. At [tex]\( (8, 0) \)[/tex]:
[tex]\[ P = 3(8) + 4(0) = 24 + 0 = 24 \][/tex]
4. At [tex]\( (0, 8) \)[/tex]:
[tex]\[ P = 3(0) + 4(8) = 0 + 32 = 32 \][/tex]
### Step 5: Identify the Maximum Value
Compare the values obtained:
- [tex]\( 8 \)[/tex]
- [tex]\( 6 \)[/tex]
- [tex]\( 24 \)[/tex]
- [tex]\( 32 \)[/tex]
The maximum value of [tex]\( P \)[/tex] is [tex]\( 32 \)[/tex] at the point [tex]\( (0, 8) \)[/tex].
### Conclusion
The solution to the optimization problem is:
- The maximum value of the objective function [tex]\( P = 3x + 4y \)[/tex] is [tex]\( 32 \)[/tex].
- This value occurs at the point [tex]\( (0, 8) \)[/tex].
1. [tex]\( x \geq 0 \)[/tex]
2. [tex]\( y \geq 0 \)[/tex]
3. [tex]\( 2x + 2y \geq 4 \)[/tex]
4. [tex]\( x + y \leq 8 \)[/tex]
we will follow these steps:
### Step 1: Rewrite the Constraints for Simplicity
- [tex]\( 2x + 2y \geq 4 \)[/tex] can be simplified to [tex]\( x + y \geq 2 \)[/tex]
- So, the constraints now are:
1. [tex]\( x \geq 0 \)[/tex]
2. [tex]\( y \geq 0 \)[/tex]
3. [tex]\( x + y \geq 2 \)[/tex]
4. [tex]\( x + y \leq 8 \)[/tex]
### Step 2: Identify the Feasible Region
We plot the constraints on the xy-plane to find the feasible region where all constraints are satisfied.
1. [tex]\( x \geq 0 \)[/tex]: This means only consider the right half of the y-axis (including the y-axis itself).
2. [tex]\( y \geq 0 \)[/tex]: This means only consider the upper half of the x-axis (including the x-axis itself).
3. [tex]\( x + y \geq 2 \)[/tex]: This is a line that extends from (2, 0) to (0, 2) and continues infinitely up to the right. The feasible region will be above this line.
4. [tex]\( x + y \leq 8 \)[/tex]: This is a line that extends from (8, 0) to (0, 8) and the feasible region will be below this line.
The feasible region is thus bounded by:
1. The line [tex]\( x + y = 2 \)[/tex]
2. The line [tex]\( x + y = 8 \)[/tex]
3. The x-axis
4. The y-axis
### Step 3: Determine the Corner Points
Find the corner points of the feasible region by determining the intersection points of the lines and axes:
1. Intersection of [tex]\( x + y = 2 \)[/tex] and [tex]\( x = 0 \)[/tex]: [tex]\( (0, 2) \)[/tex]
2. Intersection of [tex]\( x + y = 2 \)[/tex] and [tex]\( y = 0 \)[/tex]: [tex]\( (2, 0) \)[/tex]
3. Intersection of [tex]\( x + y = 8 \)[/tex] and [tex]\( x = 0 \)[/tex]: [tex]\( (0, 8) \)[/tex]
4. Intersection of [tex]\( x + y = 8 \)[/tex] and [tex]\( y = 0 \)[/tex]: [tex]\( (8, 0) \)[/tex]
5. Intersection of [tex]\( x + y = 2 \)[/tex] and [tex]\( x + y = 8 \)[/tex]: Solve the equations simultaneously:
[tex]\[ x + y = 2 \\ x + y = 8 \][/tex]
These two lines are parallel and never intersect.
The feasible region is a quadrilateral with potential corner points: [tex]\( (0, 2) \)[/tex], [tex]\( (2, 0) \)[/tex], [tex]\( (8, 0) \)[/tex], and [tex]\( (0, 8) \)[/tex].
### Step 4: Evaluate the Objective Function at Each Corner Point
Calculate the value of [tex]\( P = 3x + 4y \)[/tex] at each corner point:
1. At [tex]\( (0, 2) \)[/tex]:
[tex]\[ P = 3(0) + 4(2) = 0 + 8 = 8 \][/tex]
2. At [tex]\( (2, 0) \)[/tex]:
[tex]\[ P = 3(2) + 4(0) = 6 + 0 = 6 \][/tex]
3. At [tex]\( (8, 0) \)[/tex]:
[tex]\[ P = 3(8) + 4(0) = 24 + 0 = 24 \][/tex]
4. At [tex]\( (0, 8) \)[/tex]:
[tex]\[ P = 3(0) + 4(8) = 0 + 32 = 32 \][/tex]
### Step 5: Identify the Maximum Value
Compare the values obtained:
- [tex]\( 8 \)[/tex]
- [tex]\( 6 \)[/tex]
- [tex]\( 24 \)[/tex]
- [tex]\( 32 \)[/tex]
The maximum value of [tex]\( P \)[/tex] is [tex]\( 32 \)[/tex] at the point [tex]\( (0, 8) \)[/tex].
### Conclusion
The solution to the optimization problem is:
- The maximum value of the objective function [tex]\( P = 3x + 4y \)[/tex] is [tex]\( 32 \)[/tex].
- This value occurs at the point [tex]\( (0, 8) \)[/tex].
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
