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 $20x+15y\ge 18$
$x\ge 0,y\ge 0$
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 $₹4.6$.