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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, while each packet of the same quality of food Q contains 3 units of calcium, 20 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and almost 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?

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, while each packet of the same quality of food Q contains 3 units of calcium, 20 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and almost 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?

Grade:Upto college level

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago
Hello student,
Let x and y be the number of packets of food P and Q respectively. Obviously
x ≥ 0, y ≥ 0. Mathematical formulation of the given problem is as follows:
Minimise Z = 6x + 3y (vitamin A)
subject to the constraints
12x + 3y ≥240 (constraint on calcium), i.e. 4x + y ≥ 80 ... (1)
4x + 20y ≥460 (constraint on iron), i.e. x + 5y ≥ 115 ... (2)
6x + 4y ≤300 (constraint on cholesterol), i.e. 3x + 2y ≤ 150 ... (3)
x ≥ 0, y ≥ 0 ... (4)
Z is minimum at the point (15, 20). Hence, the amount
of vitamin A under the constraints given in the problem will be minimum, if 15 packetsof food P and 20 packets of food Q are used in the special diet. The minimum amountof vitamin A will be 150 units.
Thanks and Regards
Shaik Aasif
askIITians faculty

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