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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 atleast 240 units of calcium, atleast 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used tomaximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A?
Asked by Tarun Last Modified
Answer 
Vishaka Bhutia
Home Tutor
Let the diet contain x and y packets of foods P and Q respectively. Therefore,
x ≥ 0 and y ≥ 0
The mathematical formulation of the given problem is as follows.
Maximize z = 6x + 3y … (1)
subject to the constraints,
The feasible region determined by the system of constraints is as follows.
The corner points of the feasible region are A (15, 20), B (40, 15), and C (2, 72).
The values of z at these corner points are as follows.
| Corner point | z = 6x + 3y | |
| A(15, 20) | 150 | |
| B(40, 15) | 285 | → Maximum |
| C(2, 72) | 228 |
Thus, the maximum value of z is 285 at (40, 15).
Therefore, to maximize the amount of vitamin A in the diet, 40 packets of food P and 15 packets of food Q should be used. The maximum amount of vitamin A in the diet is 285 units.
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