Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To solve the given linear programming problem:
[tex]\[ \begin{array}{ll} \text { Maximize: } & z = 11x + 13y \\ \text { subject to: } & 7x + 5y \leq 35 \\ & 11x + y \leq 35 \\ & x \geq 0, y \geq 0 \end{array} \][/tex]
we need to follow these steps:
1. Identify the Objective Function and Constraints:
The objective function is [tex]\( z = 11x + 13y \)[/tex], which we want to maximize.
The constraints are:
[tex]\[ 7x + 5y \leq 35 \][/tex]
[tex]\[ 11x + y \leq 35 \][/tex]
[tex]\[ x \geq 0 \][/tex]
[tex]\[ y \geq 0 \][/tex]
2. Graph the Constraints:
Plot each constraint on a graph:
- For [tex]\( 7x + 5y \leq 35 \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 7 \)[/tex].
- When [tex]\( y = 0 \)[/tex], [tex]\( x = 5 \)[/tex].
- For [tex]\( 11x + y \leq 35 \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 35 \)[/tex].
- When [tex]\( y = 0 \)[/tex], [tex]\( x = \frac{35}{11} \approx 3.18 \)[/tex].
3. Find the Feasible Region:
The feasible region is where all the constraints overlap with [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex].
4. Determine the Vertices of the Feasible Region:
To find the optimal solution, we need to identify the vertices of the feasible region. The vertices are the points of intersection of the constraint lines and the x- and y-axes:
- Intersection of [tex]\( 7x + 5y = 35 \)[/tex] and [tex]\( 11x + y = 35 \)[/tex]:
- Intersection with the x-axis (where [tex]\( y = 0 \)[/tex]):
- For [tex]\( 7x + 5(0) = 35 \)[/tex], [tex]\( x = 5 \)[/tex].
- For [tex]\( 11x + 0 = 35 \)[/tex], [tex]\( x = \frac{35}{11} \approx 3.18 \)[/tex].
- Intersection with the y-axis (where [tex]\( x = 0 \)[/tex]):
- For [tex]\( 7(0) + 5y = 35 \)[/tex], [tex]\( y = 7 \)[/tex].
- For [tex]\( 11(0) + y = 35 \)[/tex], [tex]\( y = 35 \, \text{(This exceeds constraint right-hand side, so not part of feasible region)}. 5. Evaluate the Objective Function at Each Vertex: Calculate \( z = 11x + 13y \)[/tex] at each vertex to find the maximum value:
- Intersection of [tex]\( 7x + 5y = 35 \)[/tex] and [tex]\( 11x + y = 35 \)[/tex] — by solving this system of equations:
[tex]\[ 7x + 5y = 35 \][/tex]
[tex]\[ 11x + y = 35 \][/tex]
Solving these simultaneously, we find:
[tex]\[ y = 0 \Rightarrow x = 3.18 \Rightarrow z = 11(3.18) + 13(0) = 35 \][/tex]
[tex]\[ y = 7 \Rightarrow x = 0 \Rightarrow z = 11(0) + 0(7) = 0 \][/tex]
6. Optimal Solution:
The maximum value of [tex]\( z \)[/tex] is found not from x or y-axis intersections but from our feasible region straight-off calculation (as per the correct optimal vertex):
[tex]\[ \boxed{91.0} \][/tex]
Therefore, the maximum value of the objective function [tex]\( z \)[/tex] is [tex]\( \boxed{91.0} \)[/tex].
[tex]\[ \begin{array}{ll} \text { Maximize: } & z = 11x + 13y \\ \text { subject to: } & 7x + 5y \leq 35 \\ & 11x + y \leq 35 \\ & x \geq 0, y \geq 0 \end{array} \][/tex]
we need to follow these steps:
1. Identify the Objective Function and Constraints:
The objective function is [tex]\( z = 11x + 13y \)[/tex], which we want to maximize.
The constraints are:
[tex]\[ 7x + 5y \leq 35 \][/tex]
[tex]\[ 11x + y \leq 35 \][/tex]
[tex]\[ x \geq 0 \][/tex]
[tex]\[ y \geq 0 \][/tex]
2. Graph the Constraints:
Plot each constraint on a graph:
- For [tex]\( 7x + 5y \leq 35 \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 7 \)[/tex].
- When [tex]\( y = 0 \)[/tex], [tex]\( x = 5 \)[/tex].
- For [tex]\( 11x + y \leq 35 \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 35 \)[/tex].
- When [tex]\( y = 0 \)[/tex], [tex]\( x = \frac{35}{11} \approx 3.18 \)[/tex].
3. Find the Feasible Region:
The feasible region is where all the constraints overlap with [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex].
4. Determine the Vertices of the Feasible Region:
To find the optimal solution, we need to identify the vertices of the feasible region. The vertices are the points of intersection of the constraint lines and the x- and y-axes:
- Intersection of [tex]\( 7x + 5y = 35 \)[/tex] and [tex]\( 11x + y = 35 \)[/tex]:
- Intersection with the x-axis (where [tex]\( y = 0 \)[/tex]):
- For [tex]\( 7x + 5(0) = 35 \)[/tex], [tex]\( x = 5 \)[/tex].
- For [tex]\( 11x + 0 = 35 \)[/tex], [tex]\( x = \frac{35}{11} \approx 3.18 \)[/tex].
- Intersection with the y-axis (where [tex]\( x = 0 \)[/tex]):
- For [tex]\( 7(0) + 5y = 35 \)[/tex], [tex]\( y = 7 \)[/tex].
- For [tex]\( 11(0) + y = 35 \)[/tex], [tex]\( y = 35 \, \text{(This exceeds constraint right-hand side, so not part of feasible region)}. 5. Evaluate the Objective Function at Each Vertex: Calculate \( z = 11x + 13y \)[/tex] at each vertex to find the maximum value:
- Intersection of [tex]\( 7x + 5y = 35 \)[/tex] and [tex]\( 11x + y = 35 \)[/tex] — by solving this system of equations:
[tex]\[ 7x + 5y = 35 \][/tex]
[tex]\[ 11x + y = 35 \][/tex]
Solving these simultaneously, we find:
[tex]\[ y = 0 \Rightarrow x = 3.18 \Rightarrow z = 11(3.18) + 13(0) = 35 \][/tex]
[tex]\[ y = 7 \Rightarrow x = 0 \Rightarrow z = 11(0) + 0(7) = 0 \][/tex]
6. Optimal Solution:
The maximum value of [tex]\( z \)[/tex] is found not from x or y-axis intersections but from our feasible region straight-off calculation (as per the correct optimal vertex):
[tex]\[ \boxed{91.0} \][/tex]
Therefore, the maximum value of the objective function [tex]\( z \)[/tex] is [tex]\( \boxed{91.0} \)[/tex].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
