Class 12 Linear Programming · Case Study4 Markslinear_programming_models
Class 12 Maths · Linear Programming · Case Study #74
A dealer buys ceiling fans at ₹360 each and table fans at ₹240 each. Let their numbers be x,y, with x,y nonnegative integers. Storage holds at most 20 fans. For this model, take the available purchase budget as ₹5760. Profit is ₹22 per ceiling fan and ₹18 per table fan. Answer (i),(ii), and either (iii)(a) or (iii)(b).
Part (1) 1 Marks
Write the storage and purchase constraints.
Answer: x+y≤20; 360x+240y≤5760; x,y≥0 integers.
- Storage gives x+y≤20; purchase cost is 360x+240y.
- Thus 360x+240y≤5760, equivalently 3x+2y≤48, with nonnegative integer counts.
Part (2) 1 Marks
Write the profit objective.
Answer: Maximize Z=22x+18y.
- Multiply each fan count by its unit profit.
- Total profit is Z=22x+18y, to maximize.
Part (3) 2 Marks
Find the optimal numbers of fans.
Answer: 8 ceiling fans and 12 table fans.
- Relax to nonnegative real x,y and find vertices (0,0),(16,0),(8,12),(0,20).
- Profits are 0,352,392,360. The real optimum (8,12) is integral and meets both budget and storage limits.
- 8 ceiling fans and 12 table fans.
Part (4) 2 Marks
Find the maximum total profit.
Answer: ₹392.
- Relax to nonnegative real x,y and find vertices (0,0),(16,0),(8,12),(0,20).
- Profits are 0,352,392,360. The real optimum (8,12) is integral and meets both budget and storage limits.
- ₹392.