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What point in the feasible region maximizes the objective function?
Constraints:
⎧⎩⎨⎪⎪⎪⎪⎪⎪x≥0y≥0−x+3≥yy≤13x+1
Objective function: C = 5x – 4y
(3 points)
To find the point in the feasible region that maximizes the objective function, we need to find the maximum value of the objective function C = 5x - 4y.
First, let's graph the constraints in order to find the feasible region.
The constraint x ≥ 0 represents the region to the right of the y-axis.
The constraint y ≥ 0 represents the region above the x-axis.
The constraint -x + 3 ≥ y represents the region below the line with slope -1 passing through the point (3,0).
The constraint y ≤ 13x + 1 represents the region below the line with slope 13 passing through the point (0,1).
Combining all the constraints, we find that the feasible region is the triangular region bounded by the lines y = 0, x = 0, and -x + 3 = y.
Now, let's find the vertices of this feasible region.
At the intersection of y = 0 and -x + 3 = y, we have x = 3 and y = 0.
At the intersection of x = 0 and -x + 3 = y, we have x = 0 and y = 3.
At the intersection of x = 0 and y = 13x + 1, we have x = 0 and y = 1.
Therefore, the vertices of the feasible region are (3, 0), (0, 3), and (0, 1).
Now, let's calculate the objective function at each vertex.
For (3, 0):
C = 5(3) - 4(0) = 15
For (0, 3):
C = 5(0) - 4(3) = -12
For (0, 1):
C = 5(0) - 4(1) = -4
We can see that the maximum value of the objective function is 15 at the point (3, 0).
Therefore, the point (3, 0) in the feasible region maximizes the objective function.
To find the point in the feasible region that maximizes the objective function, we need to determine the corner points of the feasible region and evaluate the objective function at each of these points.
Given the following constraints:
1. x ≥ 0
2. y ≥ 0
3. -x + 3 ≥ y
4. y ≤ 13x + 1
We can start by graphing the feasible region:
Step 1: Graph the boundary lines of the first three constraints:
- x + 3 = y
- y = 0
- x = 0
Step 2: Shade the region that satisfies all the constraints. This region is the feasible region.
Step 3: Determine the corner points of the feasible region by finding the intersection points of the boundary lines.
Now we need to evaluate the objective function C = 5x - 4y at each corner point to see which point maximizes it.
Corner Point 1: (0, 0)
C = 5(0) - 4(0) = 0 - 0 = 0
Corner Point 2: (0, 3)
C = 5(0) - 4(3) = 0 - 12 = -12
Corner Point 3: (14/13, 3)
C = 5(14/13) - 4(3) = 70/13 - 12 = 70/13 - 156/13 = -86/13
Based on the evaluation of the objective function at each corner point, the point that maximizes the objective function is (0, 0) with a value of 0.
To find the point in the feasible region that maximizes the objective function, we need to evaluate the objective function at each of the vertices of the feasible region and determine which one gives the maximum value.
Step 1: Graph the feasible region:
- Start by graphing the four constraint inequalities: x ≥ 0, y ≥ 0, -x + 3 ≥ y, and y ≤ (13 - x).
- Shade the region that satisfies all four constraints. This shaded region is called the feasible region.
Step 2: Find the vertices of the feasible region:
- The vertices are the points where the boundary lines of the constraints intersect.
- In this case, the feasible region is a triangular shape with vertices at (0, 0), (4, 9), and (3, 10).
Step 3: Evaluate the objective function at each vertex:
- Substitute the x-value and y-value of each vertex into the objective function and calculate the result.
- For the vertex (0, 0):
C = 5(0) - 4(0) = 0
- For the vertex (4, 9):
C = 5(4) - 4(9) = 20 - 36 = -16
- For the vertex (3, 10):
C = 5(3) - 4(10) = 15 - 40 = -25
Step 4: Determine the maximum value of the objective function:
- From the calculations in step 3, we can see that the maximum value of the objective function occurs at the vertex (0, 0).
- Therefore, the point in the feasible region that maximizes the objective function is (0, 0).