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.
What point in the feasible region maximizes the objective function?Constraints: ⎧⎩⎨⎪⎪⎪⎪⎪⎪x≥0y≥0−x+3≥yy≤13x+1 Objective function: C = 5x – 4y
1 answer
1 answer