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A diet of a sick person must contain atleast 4000 unit of vitamins, 50 unit of proteins and 1400 calories. Two foods $\mathrm{A}$ and $\mathrm{B}$ are available at cost of $₹ 4$ and $₹ 3$ per unit respectively. If one unit of A contains 200 unit 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.
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Verified Answer
The correct answer is:
$200 x+100 y \geq 4000, x+2 y \geq 50$
$40 x+40 y \geq 1400, x \geq 0$ and $y \geq 0$
$O . F z=4 x+3 y$
$40 x+40 y \geq 1400, x \geq 0$ and $y \geq 0$
$O . F z=4 x+3 y$
$$
\begin{array}{l|c|c|c|c}
\hline \begin{array}{c}
\text { Nutrients } \\
\text { Food }
\end{array} & \begin{array}{c}
\text { Vitamins } \\
\text { (unit) }
\end{array} & \begin{array}{c}
\text { Proteins } \\
\text { (unit) }
\end{array} & \begin{array}{c}
\text { Colories } \\
\text { (unit) }
\end{array} & \begin{array}{c}
\text { Availabil } \\
\text { ity (per } \\
\text { unit) }
\end{array} \\
\hline \text { A } & 200 & 1 & 40 & 4 \text { pes } \\
\text { B } & 100 & 2 & 40 & 3 \text { pes } \\
\text { Requirement } & 4000 & 50 & 1400 & \\
\hline
\end{array}
$$
Let $z$ be the profit function and $x$ and $y$ denote the productivity of food $A$ and $B$ respectively. Then
$$
\begin{array}{c}
200 x+100 y \geq 4000 \\
x+2 y \geq 50 \\
40 x+40 \geq 1400 \\
\text { O.F } z=4 x+3 y, \quad x \geq 0 \text { and } y \geq 0
\end{array}
$$
\begin{array}{l|c|c|c|c}
\hline \begin{array}{c}
\text { Nutrients } \\
\text { Food }
\end{array} & \begin{array}{c}
\text { Vitamins } \\
\text { (unit) }
\end{array} & \begin{array}{c}
\text { Proteins } \\
\text { (unit) }
\end{array} & \begin{array}{c}
\text { Colories } \\
\text { (unit) }
\end{array} & \begin{array}{c}
\text { Availabil } \\
\text { ity (per } \\
\text { unit) }
\end{array} \\
\hline \text { A } & 200 & 1 & 40 & 4 \text { pes } \\
\text { B } & 100 & 2 & 40 & 3 \text { pes } \\
\text { Requirement } & 4000 & 50 & 1400 & \\
\hline
\end{array}
$$
Let $z$ be the profit function and $x$ and $y$ denote the productivity of food $A$ and $B$ respectively. Then
$$
\begin{array}{c}
200 x+100 y \geq 4000 \\
x+2 y \geq 50 \\
40 x+40 \geq 1400 \\
\text { O.F } z=4 x+3 y, \quad x \geq 0 \text { and } y \geq 0
\end{array}
$$
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