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Question

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)

Answer 1

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.