Class 12 Linear Programming · Long Answer3 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Long Answer #39

Maximize Z=10500*x + 9000*y subject to x + y ≤ 50; 2*x + y ≤ 80; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=495000 at (30,20).

Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (0,0), (0,50), (30,20), (40,0).
Evaluate the objective at every vertex: (0,0) → 0; (0,50) → 450000; (30,20) → 495000; (40,0) → 420000.
The region is bounded. max Z=495000 at (30,20).

[0.5 marks] Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
[1 mark] Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (0,0), (0,50), (30,20), (40,0).
[1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,50) → 450000; (30,20) → 495000; (40,0) → 420000.
[0.5 marks] The region is bounded. max Z=495000 at (30,20).

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