Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To solve the given linear programming problem, follow these step-by-step instructions:
### Step 1: Convert the Problem to Standard Form
The original problem is:
[tex]\[ \begin{array}{ll} \text{Maximize:} & z = 6x + 2y \\ \text{subject to:} & 6x - y \leq 15 \\ & 2x + y \geq 10 \\ & x \geq 2 \\ & y \leq 9 \end{array} \][/tex]
First, convert all constraints to the form [tex]\(Ax \leq b\)[/tex].
The second constraint [tex]\(2x + y \geq 10\)[/tex] can be rewritten as:
[tex]\[ -2x - y \leq -10 \][/tex]
Thus, our inequalities become:
[tex]\[ \begin{array}{ll} 6x - y \leq 15 \\ -2x - y \leq -10 \\ x \geq 2 \\ y \leq 9 \end{array} \][/tex]
### Step 2: Identify Variable Bounds
From the inequalities:
- [tex]\( x \geq 2 \)[/tex] implies a lower bound for [tex]\( x \)[/tex] of 2.
- [tex]\( y \leq 9 \)[/tex] implies an upper bound for [tex]\( y \)[/tex] of 9.
### Step 3: Solve the System of Linear Inequalities
Use linear programming techniques to find the optimal values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
First, consider the intersection of the constraints by graphing or mathematical methods.
To find the feasible region, solve the system of equations formed by the intersecting constraints:
1. Solve [tex]\( 6x - y = 15 \)[/tex] and [tex]\( -2x - y = -10 \)[/tex]:
[tex]\[ \begin{align*} 6x - y &= 15 \quad \text{(i)} \\ -2x - y &= -10 \quad \text{(ii)} \end{align*} \][/tex]
Add equation (i) and (ii):
[tex]\[ 6x - y - 2x - y = 15 - 10 \][/tex]
[tex]\[ 4x - 2y = 5 \][/tex]
[tex]\[ 4x = 20 \Rightarrow x = 4 \][/tex]
Plug [tex]\( x = 4 \)[/tex] back into one of the original equations:
[tex]\[ 6(4) - y = 15 \][/tex]
[tex]\[ 24 - y = 15 \][/tex]
[tex]\[ y = 9 \][/tex]
The intersection point is [tex]\( (4, 9) \)[/tex].
### Step 4: Evaluate the Objective Function at Feasible Corners
Considering all constraints and variable bounds, the key feasible points you need to evaluate are located at:
- [tex]\( (2, 9) \)[/tex] (from [tex]\( x \geq 2 \)[/tex] and [tex]\( y \leq 9 \)[/tex])
- [tex]\( (4, 9) \)[/tex] (from the intersection above)
Evaluate [tex]\( z = 6x + 2y \)[/tex] at these points:
- For [tex]\( (2, 9) \)[/tex]:
[tex]\[ z = 6(2) + 2(9) = 12 + 18 = 30 \][/tex]
- For [tex]\( (4, 9) \)[/tex]:
[tex]\[ z = 6(4) + 2(9) = 24 + 18 = 42 \][/tex]
### Step 5: Determine the Maximum Value
The maximum value of [tex]\( z \)[/tex] among the feasible points is [tex]\( 42 \)[/tex], achieved at [tex]\( (4, 9) \)[/tex].
Hence, the maximum value is:
[tex]\[ \boxed{42} \][/tex]
### Step 1: Convert the Problem to Standard Form
The original problem is:
[tex]\[ \begin{array}{ll} \text{Maximize:} & z = 6x + 2y \\ \text{subject to:} & 6x - y \leq 15 \\ & 2x + y \geq 10 \\ & x \geq 2 \\ & y \leq 9 \end{array} \][/tex]
First, convert all constraints to the form [tex]\(Ax \leq b\)[/tex].
The second constraint [tex]\(2x + y \geq 10\)[/tex] can be rewritten as:
[tex]\[ -2x - y \leq -10 \][/tex]
Thus, our inequalities become:
[tex]\[ \begin{array}{ll} 6x - y \leq 15 \\ -2x - y \leq -10 \\ x \geq 2 \\ y \leq 9 \end{array} \][/tex]
### Step 2: Identify Variable Bounds
From the inequalities:
- [tex]\( x \geq 2 \)[/tex] implies a lower bound for [tex]\( x \)[/tex] of 2.
- [tex]\( y \leq 9 \)[/tex] implies an upper bound for [tex]\( y \)[/tex] of 9.
### Step 3: Solve the System of Linear Inequalities
Use linear programming techniques to find the optimal values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
First, consider the intersection of the constraints by graphing or mathematical methods.
To find the feasible region, solve the system of equations formed by the intersecting constraints:
1. Solve [tex]\( 6x - y = 15 \)[/tex] and [tex]\( -2x - y = -10 \)[/tex]:
[tex]\[ \begin{align*} 6x - y &= 15 \quad \text{(i)} \\ -2x - y &= -10 \quad \text{(ii)} \end{align*} \][/tex]
Add equation (i) and (ii):
[tex]\[ 6x - y - 2x - y = 15 - 10 \][/tex]
[tex]\[ 4x - 2y = 5 \][/tex]
[tex]\[ 4x = 20 \Rightarrow x = 4 \][/tex]
Plug [tex]\( x = 4 \)[/tex] back into one of the original equations:
[tex]\[ 6(4) - y = 15 \][/tex]
[tex]\[ 24 - y = 15 \][/tex]
[tex]\[ y = 9 \][/tex]
The intersection point is [tex]\( (4, 9) \)[/tex].
### Step 4: Evaluate the Objective Function at Feasible Corners
Considering all constraints and variable bounds, the key feasible points you need to evaluate are located at:
- [tex]\( (2, 9) \)[/tex] (from [tex]\( x \geq 2 \)[/tex] and [tex]\( y \leq 9 \)[/tex])
- [tex]\( (4, 9) \)[/tex] (from the intersection above)
Evaluate [tex]\( z = 6x + 2y \)[/tex] at these points:
- For [tex]\( (2, 9) \)[/tex]:
[tex]\[ z = 6(2) + 2(9) = 12 + 18 = 30 \][/tex]
- For [tex]\( (4, 9) \)[/tex]:
[tex]\[ z = 6(4) + 2(9) = 24 + 18 = 42 \][/tex]
### Step 5: Determine the Maximum Value
The maximum value of [tex]\( z \)[/tex] among the feasible points is [tex]\( 42 \)[/tex], achieved at [tex]\( (4, 9) \)[/tex].
Hence, the maximum value is:
[tex]\[ \boxed{42} \][/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
