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Question 33 of 39

Q.Solve the following linear programming problem graphically: (Graph sheet is not required) Minimize Z = 200x + 500y subject to the constraints x+2y≥10, 3x+4y≤24, x≥0, y≥0.

West Bengal WbchseWest Bengal HS (WBCHSE) Board 2023Subjective· 5mImportance★★★★★
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Figure — This is NCERT LP Example (Minimise Z=200x+500y, x+2y>=10, 3x+4y<=24) and fig-12-3 shows exactly this minimisin
Figure — This is NCERT LP Example (Minimise Z=200x+500y, x+2y>=10, 3x+4y<=24) and fig-12-3 shows exactly this minimisin

Graph the feasible region cut out by the two constraints and the axes, list its corner (vertex) points, and evaluate ZZ at each — the minimum of a linear objective over a bounded feasible region always occurs at a vertex.

Step 1 — find the feasible region's vertices. Constraints: x+2y≥10x+2y\ge10, 3x+4y≤243x+4y\le24, x≥0, y≥0x\ge0,\ y\ge0.

On the yy-axis (x=0x=0): 2y≥10⇒y≥52y\ge10\Rightarrow y\ge5, and 4y≤24⇒y≤64y\le24\Rightarrow y\le6. So the segment from (0,5)(0,5) to (0,6)(0,6) lies on the boundary.

Intersection of the two lines x+2y=10x+2y=10 and 3x+4y=243x+4y=24: from the first, x=10−2yx=10-2y; substituting, 3(10−2y)+4y=24⇒30−2y=24⇒y=3, x=43(10-2y)+4y=24\Rightarrow30-2y=24\Rightarrow y=3,\ x=4. Vertex (4,3)(4,3).

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