Question

Kellogg is a new cereal formed of a mixture of bran and rice that contains at least $88$ grams of protein and at least $36$ milligrams of iron. Knowing that bran contains $80$ grams of proteins 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 this new cereal if bran costs $₹\, 5$ per kilogram and rice costs $₹\, 4$ per kilogram

• A. $₹\,4.8$
• B. $₹\, 4.6$
• C. $₹\,3.2$
• D. $₹\,4$

Answer 1

Answer: (B)

Let the cereal contain $x$ kg of bran and $y$ kg of rice.
Let $z$ be the cost of the new cereal. According to question,
$x$ and $y$ must satisfy the following conditions:
$x\times \frac{80}{1000} + y \times \frac{100}{1000} \ge \frac{88}{1000}$ or $20x + 25y \ge 22$
$ x\times \frac{40}{1000000} + y \times\frac{30}{1000000} \ge \frac{36}{1000000}$ or $20x +15 y \ge18$
$ x\ge0, y \ge0 $
Mathematical formulation of the $LPP$ will be
Minimize $z = 5x + 4y$
subject to constraints
$20x + 25y \ge 22, 20x + 15y \ge 18, x \ge 0, y \ge 0$
Now, we draw the lines
$ l_1 : 20x + 25y = 22$
$ l_2:20x+15y=18 $
$l_3 : x = 0, l_4: y =0$

Lines $l_1$ and $l_2$ meet at $E(0.6,0.4)$.
The feasible region has been shaded and it is an unbounded region with vertices $A, E$ and $D$.
Minimize $z = 5x + 4y$
At $A(1.1, 0) \,\,\, : z = 5 \times 1.1 + 4 \times 0 = 5.5 $
At $E(0.6, 0.4) \,\,\, : z = 5 \times 0.6 + 4 \times 0.4 = 4.6$
At $D(0,1.2) \,\,\, : z = 5 \times 0 + 4 \times 1.2 = 4.8$
$\therefore $ The minimum cost of producing this cereal is $₹\, 4.6$.