Answer :
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].
