What point in the feasible region maximizes the objective function?Constraints: ⎧⎩⎨⎪⎪⎪⎪⎪⎪x≥0y≥0−x+3≥yy≤13x+1 Objective function: C = 5x – 4y
To find the point in the feasible region that maximizes the objective function C = 5x - 4y, we need to find the maximum value of C within the feasible region.
The feasible region is defined by the following constraints:
x ≥ 0
y ≥ 0
-x + 3 ≥ y
y ≤ 13x + 1
To find the feasible region, we graph these constraints on a coordinate plane.
The first two constraints, x ≥ 0 and y ≥ 0, define the non-negative quadrant of the coordinate plane.
The constraint -x + 3 ≥ y can be rewritten as y ≤ -x + 3. When plotted on the graph, it forms a line with a slope of -1 and a y-intercept of 3. To determine which side of the line represents the feasible region, we can test a point on either side of the line. For example, plugging in the origin (0,0) into the inequality, we get 0 ≤ 3, which is true. Therefore, the feasible region is below the line -x + 3 ≥ y.
The constraint y ≤ 13x + 1 can also be rewritten as -13x + y ≤ 1. When plotted on the graph, it forms a line with a slope of 13 and a y-intercept of 1. To determine which side of the line represents the feasible region, we can test a point on either side of the line. For example, plugging in the origin (0,0) into the inequality, we get 0 ≤ 1, which is true. Therefore, the feasible region is below the line y ≤ 13x + 1.
By shading the area that satisfies all the constraints, we get a feasible region. Within this region, we need to find the point that maximizes the objective function C = 5x - 4y.
Without a graphical representation, it is difficult to determine the exact point that maximizes C = 5x - 4y. However, we can plug in some points within the feasible region to find the maximum value of C.
For example, let's plug in the coordinates (0, 0), (3, 0), and (0, 3) into the objective function C = 5x - 4y:
C(0, 0) = 5(0) - 4(0) = 0
C(3, 0) = 5(3) - 4(0) = 15
C(0, 3) = 5(0) - 4(3) = -12
Based on these calculations, the point (3, 0) within the feasible region maximizes the objective function C = 5x - 4y.
To find the point in the feasible region that maximizes the objective function, we need to perform the following steps:
1. Plot the constraints on a graph. Start by drawing the lines representing the equations of the constraints: x ≥ 0, y ≥ 0, -x + 3 ≥ y, y ≤ 13, and 3x + 1.
- The line for x ≥ 0 is a vertical line that starts from the y-axis and extends towards the positive x-direction.
- The line for y ≥ 0 is a horizontal line that starts from the x-axis and extends towards the positive y-direction.
- The line -x + 3 ≥ y is a line with a slope of -1 and a y-intercept of 3. Draw this line in a way that separates the area above it from the region below it.
- The line y ≤ 13 is a horizontal line with a y-coordinate of 13. Draw it in a way that separates the region below it from the area above it.
- The line 3x + 1 is a line with a slope of 3 and a y-intercept of 1. Draw this line in a way that separates the area above it from the region below it.
2. Identify the feasible region. The feasible region is the region that satisfies all the constraints. In this case, it is the intersection of the shaded areas that are created by the lines representing the constraints.
3. Calculate the coordinates of the vertices of the feasible region. The vertices are the points where the lines representing the constraints intersect. Each vertex represents a possible solution and will be used to evaluate the objective function.
4. Substitute the coordinates of each vertex into the objective function C = 5x - 4y to calculate the corresponding objective function values. Compare these values to determine which vertex maximizes the objective function.
5. The vertex with the highest value for the objective function is the point in the feasible region that maximizes the objective function.
By following these steps, you should be able to find the point in the feasible region that maximizes the objective function.
To find the point in the feasible region that maximizes the objective function, we need to graph the feasible region and evaluate the objective function at each corner point.
Step 1: Graph the constraints:
Constraints:
1) x ≥ 0
2) y ≥ 0
3) -x + 3 ≥ y
4) y ≤ 1/3x + 1
We can start by graphing the first three constraints on a coordinate plane:
- Plot the line x = 0 (vertical line passing through the y-axis).
- Plot the line y = 0 (horizontal line passing through the x-axis).
- Plot the line -x + 3 = y by graphing the line y = -x + 3.
Step 2: Shade the feasible region:
Since the feasible region is bounded by the inequality y ≤ 1/3x + 1 and the lines x = 0 and y = 0, shade the region below the line y = 1/3x + 1.
Step 3: Find corner points:
The corner points are the intersection points of the lines that form the feasible region.
By solving the equations, we can find the intersection points:
- Intersection of x = 0 and -x + 3 = y: (0, 3)
- Intersection of y = 0 and -x + 3 = y: (3, 0)
- Intersection of y = 0 and y = 1/3x + 1: (-3, 0)
- Intersection of -x + 3 = y and y = 1/3x + 1: (3, 2)
So, the corner points are (0, 3), (3, 0), (-3, 0), and (3, 2).
Step 4: Evaluate the objective function:
Now, we need to evaluate the objective function C = 5x - 4y at each corner point:
1) C = 5(0) - 4(3) = -12
2) C = 5(3) - 4(0) = 15
3) C = 5(-3) - 4(0) = -15
4) C = 5(3) - 4(2) = 7
Step 5: Conclusion:
From the calculations, we find that the maximum value of the objective function C = 5x - 4y is 15 at the point (3, 0). Therefore, the point (3, 0) in the feasible region maximizes the objective function.