Q93 / 150 · MHT CET 2011 · Mathematics

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.

  • Correct answer
    A
    $200 x+100 y \geq 4000, x+2 y \geq 50$ $40 x+40 y \geq 1400, x \geq 0$ and $y \geq 0$ $O . F z=4 x+3 y$
  • B
    $400 x+200 y \geq 100, x+2 y \geq 50$ $40 x+40 y \geq 1400, x \geq 0$ and $y \geq 0$ O. $F z=4 x+3 y$
  • C
    $100 x+200 y \geq 4000, x+2 y \geq 50$, $40 x+40 y \geq 1400, x \geq 0$ and $y \geq 0$ O. $F z=4 x+3 y$
  • D
    None of the above

Solution

$$ \begin{array}{l|c|c|c|c} \hline \begin{array}{c} \text { Nutrients } \\ \text { Food } \end{array} & \begin{array}{c} \text { Vitamins } \\ \text { (unit) } \end{array} & \begin{array}{c} \text { Proteins } \\ \text { (unit) } \end{array} & \begin{array}{c} \text { Colories } \\ \text { (unit) } \end{array} & \begin{array}{c} \text { Availabil } \\ \text { ity (per } \\ \text { unit) } \end{array} \\ \hline \text { A } & 200 & 1 & 40 & 4 \text { pes } \\ \text { B } & 100 & 2 & 40 & 3 \text { pes } \\ \text { Requirement } & 4000 & 50 & 1400 & \\ \hline \end{array} $$ Let $z$ be the profit function and $x$ and $y$ denote the productivity of food $A$ and $B$ respectively. Then $$ \begin{array}{c} 200 x+100 y \geq 4000 \\ x+2 y \geq 50 \\ 40 x+40 \geq 1400 \\ \text { O.F } z=4 x+3 y, \quad x \geq 0 \text { and } y \geq 0 \end{array} $$

Hence, the correct answer is option: 'A'.

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