Question

A diet of a sick person must contain at least $4000$ unit of vitamins, $50$ units of proteins and $1400$ calories. Two foods $A$ and $B$ are available at cot of Rs. $4$ and Rs. $3$ per unit respectively. If one unit of $A$ contains $200$ units of vitamins, $1$ unit of protein and $40$ calories, while one unit of food $B$ contains $100$ unit of vitamins, $2$ unit of protein and $40$ calories. Formulate the problem, so that the diet be cheapest.

• A. $\begin{array}{c}200x+100y\ge 4000,x+2y\ge 50\\ 40x+40y\ge 1400,x\ge 0\text{and}y\ge 0\\ O.Fz=4x+3y\end{array}$
• B. $\begin{array}{c}400x+200y\ge 100,x+2y\ge 50\\ 40x+40y\ge 1400,x\ge 0andy\ge 0\\ O.Fz=4x+3y\end{array}$
• C. $\begin{array}{c}100x+200y\ge 4000,x+2y\ge 50\\ 40x+40y\ge 1400,x\ge 0andy\ge 0\\ O.Fz=4x+3y\end{array}$
• D. None of the above

Answer 1

Answer: (A)

$\begin{array}{c}200x+100y\ge 4000,x+2y\ge 50\\ 40x+40y\ge 1400,x\ge 0\text{and}y\ge 0\\ O.Fz=4x+3y\end{array}$

Nutrients Food	Vitamins	Proteins	Calories	Availability
	(unit)	(unit)	(unit)	(unit)
$A$	$200$	$1$	$40$	$4$ pes
$B$	$100$	$2$	$40$	$3$ pes
Requirement	$4000$	$50$	$1400$

Let $z$ be the profit function and $x$ and $y$ denote the productivity of food $A$ and $B$ respectively.
Then
$200x+100y\ge 4000$
$x+2y\ge 50$
$40x+40\ge 1400$
O.F $z=4x+3y$, $x\ge 0$ and $y\ge 0$