Question

The point at which the maximum value of x + y, subject to the constraints x + 2y ≤ 70, 2x + y ≤ 95, x, y ≥ 0 is obtained, is

A. (30, 25)

B. (20, 35)

C. (35, 20)

D. (40, 15)

Answer 1

Given objective function is Z = x + y

Constraints are:

x + 2y ≤ 70

2x + y ≤ 95

x, y ≥ 0

Let us consider these constraints as equations for a while, then we will have,

x + 2y = 70 ----- (1)

2x + y = 95 ----- (2)

Now, graph the equations, by transforming the equations to intercept form of line.

Equation (1) dividing throughout by 70

The line x + 2y = 70 can be plot in the graph as a line passing through the points, (70, 0) and (0, 35) as 70 and 35 are the intercepts of the line on the x-axis and y-axis respectively.

Similarly equation (2) can be divided 95 to get

The line 2x + y = 95 can be plot in the graph as a line passing through the points, \(\left(\frac{95}{2},0 \right)\) and (0, 95) as \(\frac{95}{2}\) and 95 are the intercepts of the line on the x-axis and y-axis respectively.

By considering the constraints x , y ≥ 0, this clearly shows that the region can only be in the 1^st quadrant. The graph of the inequations will look like,

The points OABC is the feasible region of the LPP.

Now from the points O, A , B and C the vertices of the polygon formed by the constraints, one of the points will provide the maximum solution to the function Z = x + y

Now checking the points, O, A, B and C by substituting in Z = x + y.

From the above values, it is clear that Z maximized at point B( 40,15).