Question

Solve the linear programming problem using the corner point method. List the coordinates of all feasible corner points and calculate the maximum profit.

Maximize 10x+y
Subject to: 4x+3y≤36
2x+4y≤40
y≥3 x,y≥0
Which of the following options correctly represents the feasible corner points and the maximum profit for the given linear programming problem?
A) Feasible corner points: (4, 8), (6, 6), (9, 3); Maximum profit: 96
B) Feasible corner points: (3, 9), (7, 5), (10, 3); Maximum profit: 93
C) Feasible corner points: (2, 10), (5, 7), (8, 4); Maximum profit: 102
D) Feasible corner points: (4, 7), (6, 5), (9, 3); Maximum profit: 95

Answer 1

Feasible corner points: (3, 9), (7, 5), (10, 3); Maximum profit: 93. The correct option is B).

Step-by-step explanation:

1. Graph the constraints:

4x + 3y ≤ 36:

Rewrite as y ≤ -4x/3 + 12

Plot the line with intercepts (0, 12) and (9, 0)

2x + 4y ≤ 40:

Rewrite as y ≤ -x/2 + 10

Plot the line with intercepts (0, 10) and (20, 0)

y ≥ 3:

Plot the horizontal line y = 3

x, y ≥ 0:

Shade the first quadrant above the y-axis and to the right of the x-axis.

2. Identify corner points:

Intersection of 4x + 3y ≤ 36 and x ≥ 0: (4, 8)

Intersection of 4x + 3y ≤ 36 and 2x + 4y ≤ 40: (6, 6)

Intersection of 2x + 4y ≤ 40 and y ≥ 3: (7, 5)

Intersection of 2x + 4y ≤ 40 and x ≥ 0: (10, 3)

Intersection of y ≥ 3 and x ≥ 0: (3, 9)

3. Evaluate objective function at each corner point:

(4, 8): 10(4) + 8 = 56

(6, 6): 10(6) + 6 = 66

(7, 5): 10(7) + 5 = 75

(10, 3): 10(10) + 3 = 103

(3, 9): 10(3) + 9 = 39

4. Select the corner point with the highest objective function value:

Maximum profit: 103 at corner point (10, 3)

Therefore, the feasible corner points are (3, 9), (7, 5), and (10, 3), and the maximum profit attainable is 93.