Q13 of 118 Page 30

A farmer mixes two brands P and Q of cattle feed. Brand P, costing ₹ 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing ₹ 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

The above information can be expressed with the help of the following table:



Let ‘x’ bags of P and ‘y’ bags of Q be bought.


Cost of P = 250x


Cost of Q = 200y


Cost of mixture = 250x + 200y


Now,


⟹ 3x + 1.5y ≥18


i.e. the minimum requirement of element A from both P and Q combined is 18units, each of which has 3units and 1.5units of element A.


⟹ 2.5x + 11.25y ≥ 45


i.e. the minimum value of element B from both P and Q combined is 45units, each of which contains 2.5units and 11,25units of element B.


⟹ 2x + 3y ≥ 24


i.e. the minimum value of element C from both P and Q combined is 24units, each of which contains 2units and 3 units of element C.


Hence, mathematical formulation of the above LPP is as follows :


Find ‘x’ and ‘y’ that:


Minimises Z = 250x + 200y


Subject to the following constraints:


(i) 3x + 1.5y ≥18


(ii) 2.5x + 11.25y ≥ 45


(iii) 2x + 3y ≥ 24


(iv) x,y ≥0 (∵ quantity cant be negative)



The feasible region is unbounded


The corner points of the feasible region are as follows:



Z is minimised at B(3,6) i.e. 3 bags of P and 6 bags of Q should be purchased to achieve the minimum cost of the mixture per bag.


The minimum cost of the mixture is ₹1950.


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A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

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A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below :


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A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.


If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?