question_answer 1)

(Diet problem) A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A, while 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 maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A ? 

Answer:

Let number of packets of food P = x           and number of packets of food Q = y                            The contents of each packet of food P and Q is given as

Food	Calcium	Iron	Cholesterol	Vitamin A
P	12	4	6	6
Q	3	20	4	3
Minimum requirement	240	460	At most 300

      The above L.P.P. is given as       Maximize, Z = 6x + 3y, subject to the constraints 12x + 3y  240, 4x + 20y  460             6x + 4y  300, x, y       i.e., 4x + y       3x + 2y       L1 : 4x + y = 80                    L2 : x + 5y = 115                                   L3 : 3x + 2y = 150                                           Here Z is manimum at point G(40, 15)       Hence amount of vitamin A wil be maximum if 40 packets of food P and 15 packets of food Q are to be used.