CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Linear Programming

Q.8. Two tailors A and B are paid Rs150 and Rs200 per day respectively. A can stich 6 shirts and 4 paints while B can stich 10 shirts and 4 paints per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 paints. Solve the problem graphically.

Solution :

Do yourself. [Ans. = Tailor A works for 5 days and B for 3 days;
Minimum cost = Rs1,350]

Q.9. A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs10,500 and Rs9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, not more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximize the total profit of the society ?

Solution :

Let x hectare of land be allocatedto crop X and y hectare to crop Y.
Clearly x ≥ 0 and y ≥ 0.
Profit per hectare on crop X = Rs10,500,
Profit per hectare on crop Y = Rs9,000,
Therefore total profit = Rs(10,500 x + 9,000 y).
The mathematical formulation of the problem is as follows :
Maximize Z = 10,500 x + 9,000 y
subject to the constraints :
x + y ≤ 50 (constraint related to land) ----------------- (1)
20 x + 10 y ≤ 800 (constraint related to use of herbicide)
Or, 2 x + y ≤ 80 ----------------------------------------- (2)
And x ≥ 0, y ≥ 0 (non negative constraint) -------------------- (3)

Fig.

In the graph we see that the feasible region is bounded. The corner points are O (0, 0), A (40, 0), B (30, 20) and C (0, 50). Let us evaluate the objective function Z = 10,500 x + 9,000 y at these vertices to find which one gives the maximum profit.

Hence , society will get the maximum profit of Rs4,95,000 by allocating 30 hectare for crop X and 20 hectare for crop Y. [Ans.]

Paper By Mr. M.P.Keshari
Email Id : [email protected]
Ph No. : 09434150289