MCQ

The maximum value of *Z* = 4*x* + 3*y* subjected to the constraints 3*x* + 2*y* ≥ 160, 5*x* + 2*y* ≥ 200, *x* + 2*y* ≥ 80; *x*, *y* ≥ 0 is

#### Options

320

300

230

none of these

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#### Solution

none of these

We need to maximize the function Z = 4*x* + 3*y*

Converting the given inequations into equations, we obtain

\[3x + 2y = 160, 5x + 2y = 200, x + 2y = 80, x = 0 \text{ and } y = 0\]

Region represented by 3*x** *+ 2*y** *≥ 160:

The line 3*x** *+ 2*y** *= 160 meets the coordinate axes at \[A\left( \frac{160}{3}, 0 \right)\] and *B*(0, 80) respectively. By joining these points we obtain the line 3*x** *+ 2*y** *= 160.Clearly (0,0) does not satisfies the inequation 3*x** *+ 2*y** *≥ 160. So,the region in *xy* plane which does not contain the origin represents the solution set of the inequation 3*x** *+ 2*y** *≥ 160.

Region represented by 5

*x*

*+2*

*y*≥ 200 :

The line 5

*x*

*+2*

*y*= 200 meets the coordinate axes at

*C*(40,0) and

*D*(0, 100) respectively. By joining these points we obtain the line 5

*x*

*+2*

*y*= 200.Clearly (0,0) does not satisfies the inequation 5

*x*

*+2*

*y*≥ 200. So,the region which does not contain the origin represents the solution set of the inequation 5

*x*

*+2*

*y*≥ 200.

Region represented by

*x*

*+2*

*y*≥ 80:

The line

*x*

*+2*

*y*= 80 meets the coordinate axes at

*E*(80,0) and

*F*(0, 40) respectively. By joining these points we obtain the line

*x*

*+2*

*y*= 80.Clearly (0,0) does not satisfies the inequation

*x*

*+2*

*y*≥ 80. So,the region which does not contain the origin represents the solution set of the inequation

*x*+2

*y*≥ 80.

Region represented by

Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations

The feasible region determined by the system of constraints 3

*x*≥ 0 and*y*≥ 0:Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations

*x*≥ 0, and*y*≥ 0.The feasible region determined by the system of constraints 3

*x**+ 2**y**≥ 160, 5**x**+2**y*≥ 200,*x**+2**y*≥ 80,*x*≥ 0, and*y*≥ 0 are as follows.Here, we see that the feasible region is unbounded. Therefore,maximum value is infinity.

Concept: Introduction of Linear Programming

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