# A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/Kg of vitamin C. It costs Rs  50 per kg to produce food I. Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C and it costs Rs 70 per kg to produce food II. Formulate this problem as a LPP to minimise the cost of a mixture that will produce the required diet. Also, find the minimum cost.

Let the dietician mix x Kg of food I  & y Kg of food II.
To minimize: $z= Rs\left ( 50x+70y \right )$
subject to constants:
$2x+y\geq 8,\; x+2y\geq 10,\: x,y\geq 0$
corner points                                   value of  $Z\left ( in\: Rs \right )$
$A\left ( 10,0 \right )$                                        500
$B\left ( 2,4 \right )$                                          380 $\leftarrow$ minimum
$C\left ( 0,8 \right )$                                           560
since the feasible region is bounded. so 380 may or may not be the minimum value of Z. To check,Let $z< 380\: i.e$
$50x+70y< 380\; \; i.e\; \; 5x+7y< 38$
Note that the feasible region and $5x+7y< 38$   do not have any common point
$\therefore mim^{th}$  value of z is Rs 380 when x= amount of food I=2 Kg and y=amount of food II=4 Kg

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