Question

A new cereal, formed of a mixture of bran and rice, contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing a kilogram of this new cereal if bran costs? ₹ 28 per kilogram and rice costs ₹ 25 per kilogram.

Answer 1

Let x kg of bran and y kg of rice be contained in a cereal.

∴ Minimize, z = 28x + 25y

Subject to constraints

x ≥ 0, y ≥ 0

First we draw the line AB and CD whose equation are 20x + 25y = 22 and 20x + 15 y = 18 respectively.

Now,

The feasible region is XBPCY which is shaded in the figure. 

The vertices of the feasible region are B (1.1, 0), P and C (0, 1.2). P is the point of intersection of the lines 20x + 25y = 22 and 20x + 15y= 18. Solving those we get the point P (0.6,0.4)

The minimum value of z is :

Hence, the minimum value of z is 26.8 at the point P(0.6, 0.4)