Solve the linear programming problem by the method of corners. Maximize subject to
A) x = 25, y = 5, P = 420
B) x = 15, y = 17.5, P = 165
C) x = 0, y = 25, P = 150
D) x = 0, y = 17.5, P = 130

Question

Solve the linear programming problem by the method of corners. Maximize  subject to     
A) x = 25, y = 5, P = 420 
B) x = 15, y = 17.5, P = 165 
C) x = 0, y = 25, P = 150 
D) x = 0, y = 17.5, P = 130

Quizplus · Accepted Answer

This problem can be solved by graphing the constraints and finding the corners or vertices of the feasible region. The feasible region is the region where all constraints are satisfied.

The constraints are:
- x + y ≤ 25 
- 2x + y ≤ 37.5 
- x ≤ 15 
- y ≤ 17.5 
- x, y ≥ 0

Plotting these on a graph, we get:

(0,17.5)           (15,10)
            *                     *
               *                *
                    *      *
                        * 
                        (0,0)               (25,0)

The feasible region is the shaded region bounded by the lines and the axes. The corners or vertices of the feasible region are:
- (0, 0)
- (15, 10)
- (12.5, 12.5)
- (0, 17.5)

To maximize the objective function P = 11x + 17y, we evaluate P at each corner:
- P(0, 0) = 0
- P(15, 10) = 11(15) + 17(10) = 245
- P(12.5, 12.5) = 11(12.5) + 17(12.5) = 297.5
- P(0, 17.5) = 17(17.5) = 297.5

The maximum value of P is 297.5, which occurs at corners (12.5, 12.5) and (0, 17.5). However, (0, 17.5) violates the constraint x ≤ 15. Therefore, the optimal solution is at (15, 10) with P = 245.

Thus, the answer is B) x = 15, y = 17.5, P = 245.

Solve the Linear Programming Problem by the Method of Corners