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Question: Find the point at which the maximum value of P = 10x + 5y subject to the constraints x ≥ 0, y ≥ 0 and 3x + 4y ≤ 12 occurs.


A ) (4, 0)
B ) (0, 0)
C ) (0, 3)
D ) (1, 1)

Steps to derive

1 Objective function is P = 10x + 5y

2 Constraints are x ≥ 0, y ≥ 0 and 3x + 4y ≤ 12.

3 The feasible region determined by the constraints is shown.
 [Draw the constraints.]


4 From the graph the vertices are (0, 0), (4, 0) and (0, 3).

5 To evaluate the minimum and maximum values of P, we evaluate P = 10x + 5y at each of the vertices.

6 At (0, 0), P = 10(0) + 5(0) = 0
 [Substitute the values.]

7 At (4, 0), P = 10(4) + 5(0) = 40
 [Substitute the values.]

8 At (0, 3), P = 10(0) + 5(3) = 15
 [Substitute the values.]

9 So, the maximum value of P is 40 which occurs when x = 4 and y = 0 .



Hence the right answer is Option A

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