Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To solve this problem, let’s follow these steps:
1. Graph the constraints: We will graph the feasible region determined by the constraints.
2. Identify the corner points: Determine the intersection points of the constraints.
3. Evaluate the objective function at the corner points: Calculate the value of the objective function [tex]\(C = 7x - 3y\)[/tex] at each corner point.
4. Determine the maximum value: Identify which point gives the maximum value for the objective function [tex]\(C\)[/tex].
### Step 1: Graph the Constraints
The given constraints are:
1. [tex]\(x \geq 0\)[/tex]
2. [tex]\(y \geq 0\)[/tex]
3. [tex]\(y \leq \frac{1}{5}x + 2\)[/tex]
4. [tex]\(y + x \leq 5\)[/tex]
### Step 2: Identify the Intersection Points
To find the intersection points, we solve the equations derived from the constraints:
#### Intersection of [tex]\( y = \frac{1}{5}x + 2 \)[/tex] and [tex]\( y + x = 5 \)[/tex]:
Set [tex]\(y = \frac{1}{5}x + 2\)[/tex] into [tex]\( y + x = 5\)[/tex]:
[tex]\[ \frac{1}{5}x + 2 + x = 5 \][/tex]
[tex]\[ x + \frac{1}{5}x = 3 \][/tex]
[tex]\[ \frac{6}{5}x = 3 \][/tex]
[tex]\[ x = \frac{5 \cdot 3}{6} = 2.5 \][/tex]
[tex]\[ y = \frac{1}{5}(2.5) + 2 = 2.5 \][/tex]
So, one of the intersection points is [tex]\((2.5, 2.5)\)[/tex].
#### Other points from constraints:
- [tex]\( (0, 2) \)[/tex] : Intersection of [tex]\( y = \frac{1}{5}x + 2 \)[/tex] and [tex]\(x = 0 \)[/tex]
- [tex]\( (5, 0) \)[/tex] : Intersection of [tex]\( y + x = 5 \)[/tex] and [tex]\(y = 0 \)[/tex]
- [tex]\( (0, 0) \)[/tex] : Intersection of [tex]\( y = 0 \)[/tex] and [tex]\(x = 0 \)[/tex]
So, the corner points of the feasible region are:
[tex]\[ (2.5, 2.5), (0, 2), (0, 0), (5, 0) \][/tex]
### Step 3: Evaluate the Objective Function at Each Corner Point
We now calculate the value of the objective function [tex]\(C = 7x - 3y\)[/tex] at each corner point:
- At [tex]\((2.5, 2.5)\)[/tex]:
[tex]\[ C = 7(2.5) - 3(2.5) = 17.5 - 7.5 = 10 \][/tex]
- At [tex]\((0, 2)\)[/tex]:
[tex]\[ C = 7(0) - 3(2) = 0 - 6 = -6 \][/tex]
- At [tex]\((0, 0)\)[/tex]:
[tex]\[ C = 7(0) - 3(0) = 0 \][/tex]
- At [tex]\((5, 0)\)[/tex]:
[tex]\[ C = 7(5) - 3(0) = 35 - 0 = 35 \][/tex]
### Step 4: Determine the Maximum Value
Comparing the values obtained:
[tex]\[ 10, -6, 0, 35 \][/tex]
The maximum value of the objective function is 35 which occurs at the point [tex]\((5, 0)\)[/tex].
### Conclusion
Therefore, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that maximize the objective function [tex]\(C = 7x - 3y\)[/tex] within the given constraints are:
[tex]\[ x = 5 \quad \text{and} \quad y = 0 \][/tex]
Thus, the maximum value of the objective function is 35 at the point [tex]\((5, 0)\)[/tex].
1. Graph the constraints: We will graph the feasible region determined by the constraints.
2. Identify the corner points: Determine the intersection points of the constraints.
3. Evaluate the objective function at the corner points: Calculate the value of the objective function [tex]\(C = 7x - 3y\)[/tex] at each corner point.
4. Determine the maximum value: Identify which point gives the maximum value for the objective function [tex]\(C\)[/tex].
### Step 1: Graph the Constraints
The given constraints are:
1. [tex]\(x \geq 0\)[/tex]
2. [tex]\(y \geq 0\)[/tex]
3. [tex]\(y \leq \frac{1}{5}x + 2\)[/tex]
4. [tex]\(y + x \leq 5\)[/tex]
### Step 2: Identify the Intersection Points
To find the intersection points, we solve the equations derived from the constraints:
#### Intersection of [tex]\( y = \frac{1}{5}x + 2 \)[/tex] and [tex]\( y + x = 5 \)[/tex]:
Set [tex]\(y = \frac{1}{5}x + 2\)[/tex] into [tex]\( y + x = 5\)[/tex]:
[tex]\[ \frac{1}{5}x + 2 + x = 5 \][/tex]
[tex]\[ x + \frac{1}{5}x = 3 \][/tex]
[tex]\[ \frac{6}{5}x = 3 \][/tex]
[tex]\[ x = \frac{5 \cdot 3}{6} = 2.5 \][/tex]
[tex]\[ y = \frac{1}{5}(2.5) + 2 = 2.5 \][/tex]
So, one of the intersection points is [tex]\((2.5, 2.5)\)[/tex].
#### Other points from constraints:
- [tex]\( (0, 2) \)[/tex] : Intersection of [tex]\( y = \frac{1}{5}x + 2 \)[/tex] and [tex]\(x = 0 \)[/tex]
- [tex]\( (5, 0) \)[/tex] : Intersection of [tex]\( y + x = 5 \)[/tex] and [tex]\(y = 0 \)[/tex]
- [tex]\( (0, 0) \)[/tex] : Intersection of [tex]\( y = 0 \)[/tex] and [tex]\(x = 0 \)[/tex]
So, the corner points of the feasible region are:
[tex]\[ (2.5, 2.5), (0, 2), (0, 0), (5, 0) \][/tex]
### Step 3: Evaluate the Objective Function at Each Corner Point
We now calculate the value of the objective function [tex]\(C = 7x - 3y\)[/tex] at each corner point:
- At [tex]\((2.5, 2.5)\)[/tex]:
[tex]\[ C = 7(2.5) - 3(2.5) = 17.5 - 7.5 = 10 \][/tex]
- At [tex]\((0, 2)\)[/tex]:
[tex]\[ C = 7(0) - 3(2) = 0 - 6 = -6 \][/tex]
- At [tex]\((0, 0)\)[/tex]:
[tex]\[ C = 7(0) - 3(0) = 0 \][/tex]
- At [tex]\((5, 0)\)[/tex]:
[tex]\[ C = 7(5) - 3(0) = 35 - 0 = 35 \][/tex]
### Step 4: Determine the Maximum Value
Comparing the values obtained:
[tex]\[ 10, -6, 0, 35 \][/tex]
The maximum value of the objective function is 35 which occurs at the point [tex]\((5, 0)\)[/tex].
### Conclusion
Therefore, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that maximize the objective function [tex]\(C = 7x - 3y\)[/tex] within the given constraints are:
[tex]\[ x = 5 \quad \text{and} \quad y = 0 \][/tex]
Thus, the maximum value of the objective function is 35 at the point [tex]\((5, 0)\)[/tex].
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.