Question

38. A Linear Programming Problem is as follows: Maximise / Minimise objective function Z = 2x – y + 5 Subject to the constraints 3x + 4y = 60 x + 3y S 30 *20, y20 If the corner points of the feasible region are A (0, 10), B(12, 6), C(20, 0) and 0(0,0), then which of the following is true ? (a) Maximum value of Z is 40 5) Minimum value of Z is - 5 (c) Difference of maximum and minimum values of Z is 35 Z (d) At two corner points, value of Z are equal 1996 0964​

Answer 1

Given Question :-

A Linear Programming Problem is as follows:

Maximise / Minimise objective function Z = 2x – y + 5

Subject to the constraints

$\rm \: 3x + 4y \leqslant 60$

$\rm \: x + 3y \leqslant 30$

$\rm \: x \geqslant 0$

$\rm \: y \geqslant 0$

If the corner points of the feasible region are A (0, 10), B(12, 6), C(20, 0) and 0(0,0), then which of the following is true ?

(a) Maximum value of Z is 40

b) Minimum value of Z is - 5

(c) Difference of maximum and minimum values of Z is 35

(d) At two corner points, value of Z are equal.

$\large\underline{\sf{Solution-}}$

Given objective function is

Maximise or Minimise Z = 2x - y + 5

subject to the constraints

$\rm \: 3x + 4y \leqslant 60$

$\rm \: x + 3y \leqslant 30$

$\rm \: x \geqslant 0$

$\rm \: y \geqslant 0$

Now, Corner points of the feasible region are

$\rm \: A \: is \: (0, \: 10)$

$\rm \: B \: is \: (12, \: 6)$

$\rm \: C \: is \: (20, \: 0)$

$\rm \: O \: is \: (0, \: 0)$

Now, Value of Z at corner points are given below :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf Corner\:point & \bf Value\:of\:Z = 2x - y + 5 \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf A(0,10) & \sf - 5 \\ \\ \sf B(12,6) & \sf 23 \\ \\ \sf C(20,0) & \sf 45\\ \\ \sf O(0,0) & \sf 0 \end{array}} \\ \end{gathered}$

So, from above table, we concluded that

Minimum value of Z is - 5 at A (0, 10).

Maximum value of Z is 45 at C (20, 0).

The difference between Maximum value and Minimum value is 50.

$\rm\implies \:Option \: (b) \: is \: correct$