Question

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 in the diet?

Answer 1

Let the diet contain x and y packets of food 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,
4x+y≥80...(2)
x+5y≥115...(3)
3x+2y≤150...(4)
x,y≥0...(5)

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| $\rightarrow$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.