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Question 29 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 2019Subjective· 5mImportance★★★★★
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Figure — Draw the first-quadrant LPP feasible region for x+2y>=10, 3x+4y<=24
Figure — Draw the first-quadrant LPP feasible region for x+2y>=10, 3x+4y<=24

Find the feasible region's corner points and evaluate ZZ at each; the smallest value is the minimum (region is bounded).

Constraints: x+2y≥10x+2y\ge10, 3x+4y≤243x+4y\le24, x≥0,y≥0x\ge0,y\ge0.

Boundary lines: x+2y=10x+2y=10 passes through (10,0)(10,0) and (0,5)(0,5). 3x+4y=243x+4y=24 passes through (8,0)(8,0) and (0,6)(0,6).

Intersection of the two lines: From x+2y=10x+2y=10, x=10−2yx=10-2y. Substitute into 3x+4y=243x+4y=24: 3(10−2y)+4y=24⇒30−6y+4y=24⇒−2y=−6⇒y=3, x=43(10-2y)+4y=24 \Rightarrow 30-6y+4y=24 \Rightarrow -2y=-6 \Rightarrow y=3,\ x=4. Intersection: (4,3)(4,3).

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