Class 12 Linear Programming · Very Long Answer5 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Very Long Answer #67

Maximize Z=7*x + 10*y subject to 4*x + 6*y ≤ 240; 6*x + 3*y ≤ 240; -x ≤ -10; -x ≤ 0; -y ≤ 0. Solve graphically.

✅ Step-by-Step Solution

Final Answer: max Z=410 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.
Boundary equations: 4*x + 6*y = 240; 6*x + 3*y = 240; -x = -10; -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: (10,0), (10,100/3), (30,20), (40,0).
Evaluate the objective at every vertex: (10,0) → 70; (10,100/3) → 1210/3; (30,20) → 410; (40,0) → 280.
The region is bounded. max Z=410 at (30,20).

[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: 4*x + 6*y = 240; 6*x + 3*y = 240; -x = -10; -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: (10,0), (10,100/3), (30,20), (40,0).
[1 mark] Evaluate the objective at every vertex: (10,0) → 70; (10,100/3) → 1210/3; (30,20) → 410; (40,0) → 280.
[1 mark] The region is bounded. max Z=410 at (30,20).

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