Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #44

Maximize Z=1000*x + 500*y subject to 3*x + 5*y ≤ 225; 2*x + y ≤ 80; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=40000 at (25,30), (40,0) and every point on their joining segment.

Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
Boundary equations: 3*x + 5*y = 225; 2*x + y = 80; -x = 0; -y = 0. Use an off-line test point to choose each half-plane.
Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (0,0), (0,45), (25,30), (40,0).
Evaluate the objective at every vertex: (0,0) → 0; (0,45) → 22500; (25,30) → 40000; (40,0) → 40000.
The region is bounded. max Z=40000 at (25,30), (40,0) and every point on their joining segment.

[1 mark] Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
[1 mark] Boundary equations: 3*x + 5*y = 225; 2*x + y = 80; -x = 0; -y = 0. Use an off-line test point to choose each half-plane.
[1 mark] Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (0,0), (0,45), (25,30), (40,0).
[1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,45) → 22500; (25,30) → 40000; (40,0) → 40000.
[1 mark] The region is bounded. max Z=40000 at (25,30), (40,0) and every point on their joining segment.

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Linear Programming.