Get comprehensive solutions to your problems with IDNLearn.com. Get comprehensive and trustworthy answers to all your questions from our knowledgeable community members.
Sagot :
To maximize the objective function [tex]\( z = 5x + 3y \)[/tex] given the constraints:
1. [tex]\( x + 4y \leq 28 \)[/tex]
2. [tex]\( 6x + y \leq 30 \)[/tex]
3. [tex]\( x \geq 0 \)[/tex]
4. [tex]\( y \geq 0 \)[/tex]
let’s follow the steps to solve this linear programming problem:
### Step 1: Identify the Constraints
The constraints define the feasible region. They are:
- [tex]\( x + 4y \leq 28 \)[/tex]
- [tex]\( 6x + y \leq 30 \)[/tex]
- [tex]\( x \geq 0 \)[/tex]
- [tex]\( y \geq 0 \)[/tex]
### Step 2: Graph the Constraints
To graph the constraints in a two-dimensional plane:
1. [tex]\( x + 4y = 28 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 7 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = 28 \)[/tex]
2. [tex]\( 6x + y = 30 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 30 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = 5 \)[/tex]
These lines, along with the non-negativity constraints [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex], define a polygonal feasible region on the xy-plane.
### Step 3: Determine the Feasible Region
The feasible region is bounded by the lines [tex]\( x + 4y = 28 \)[/tex], [tex]\( 6x + y = 30 \)[/tex], and the coordinate axes [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex]. This region is a convex polygon.
### Step 4: Identify the Corner Points
The vertices (corner points) of the feasible region are found by solving the system of linear equations given by the intersections of the constraints:
1. Intersection of [tex]\( x + 4y = 28 \)[/tex] and [tex]\( 6x + y = 30 \)[/tex]
2. Intersection of [tex]\( x + 4y = 28 \)[/tex] and [tex]\( y = 0 \)[/tex]
3. Intersection of [tex]\( 6x + y = 30 \)[/tex] and [tex]\( x = 0 \)[/tex]
4. The origin (0,0)
By solving these intersections, the vertices (corner points) are determined. These are:
1. [tex]\( (4, 6) \)[/tex]
2. [tex]\( (0, 7) \)[/tex]
3. [tex]\( (5, 0) \)[/tex]
### Step 5: Evaluate the Objective Function at Each Corner Point
Evaluate [tex]\( z = 5x + 3y \)[/tex] at each corner point:
1. At [tex]\( (4, 6) \)[/tex]:
- [tex]\( z = 5(4) + 3(6) = 20 + 18 = 38 \)[/tex]
2. At [tex]\( (0, 7) \)[/tex]:
- [tex]\( z = 5(0) + 3(7) = 0 + 21 = 21 \)[/tex]
3. At [tex]\( (5, 0) \)[/tex]:
- [tex]\( z = 5(5) + 3(0) = 25 + 0 = 25 \)[/tex]
### Step 6: Identify the Maximum Value
The maximum value of [tex]\( z \)[/tex] is obtained at the point where [tex]\( z = 38 \)[/tex].
### Conclusion
The maximum value of [tex]\( z \)[/tex] is [tex]\( 38 \)[/tex] at the point:
[tex]\[ \begin{array}{l} x = 4 \\ y = 6 \end{array} \][/tex]
So the final answer is:
Maximum is [tex]\( 38 \)[/tex] at
[tex]\[ \begin{array}{l} x = 4 \\ y = 6 \end{array} \][/tex]
1. [tex]\( x + 4y \leq 28 \)[/tex]
2. [tex]\( 6x + y \leq 30 \)[/tex]
3. [tex]\( x \geq 0 \)[/tex]
4. [tex]\( y \geq 0 \)[/tex]
let’s follow the steps to solve this linear programming problem:
### Step 1: Identify the Constraints
The constraints define the feasible region. They are:
- [tex]\( x + 4y \leq 28 \)[/tex]
- [tex]\( 6x + y \leq 30 \)[/tex]
- [tex]\( x \geq 0 \)[/tex]
- [tex]\( y \geq 0 \)[/tex]
### Step 2: Graph the Constraints
To graph the constraints in a two-dimensional plane:
1. [tex]\( x + 4y = 28 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 7 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = 28 \)[/tex]
2. [tex]\( 6x + y = 30 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 30 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = 5 \)[/tex]
These lines, along with the non-negativity constraints [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex], define a polygonal feasible region on the xy-plane.
### Step 3: Determine the Feasible Region
The feasible region is bounded by the lines [tex]\( x + 4y = 28 \)[/tex], [tex]\( 6x + y = 30 \)[/tex], and the coordinate axes [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex]. This region is a convex polygon.
### Step 4: Identify the Corner Points
The vertices (corner points) of the feasible region are found by solving the system of linear equations given by the intersections of the constraints:
1. Intersection of [tex]\( x + 4y = 28 \)[/tex] and [tex]\( 6x + y = 30 \)[/tex]
2. Intersection of [tex]\( x + 4y = 28 \)[/tex] and [tex]\( y = 0 \)[/tex]
3. Intersection of [tex]\( 6x + y = 30 \)[/tex] and [tex]\( x = 0 \)[/tex]
4. The origin (0,0)
By solving these intersections, the vertices (corner points) are determined. These are:
1. [tex]\( (4, 6) \)[/tex]
2. [tex]\( (0, 7) \)[/tex]
3. [tex]\( (5, 0) \)[/tex]
### Step 5: Evaluate the Objective Function at Each Corner Point
Evaluate [tex]\( z = 5x + 3y \)[/tex] at each corner point:
1. At [tex]\( (4, 6) \)[/tex]:
- [tex]\( z = 5(4) + 3(6) = 20 + 18 = 38 \)[/tex]
2. At [tex]\( (0, 7) \)[/tex]:
- [tex]\( z = 5(0) + 3(7) = 0 + 21 = 21 \)[/tex]
3. At [tex]\( (5, 0) \)[/tex]:
- [tex]\( z = 5(5) + 3(0) = 25 + 0 = 25 \)[/tex]
### Step 6: Identify the Maximum Value
The maximum value of [tex]\( z \)[/tex] is obtained at the point where [tex]\( z = 38 \)[/tex].
### Conclusion
The maximum value of [tex]\( z \)[/tex] is [tex]\( 38 \)[/tex] at the point:
[tex]\[ \begin{array}{l} x = 4 \\ y = 6 \end{array} \][/tex]
So the final answer is:
Maximum is [tex]\( 38 \)[/tex] at
[tex]\[ \begin{array}{l} x = 4 \\ y = 6 \end{array} \][/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.
