question_answer

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains atleast 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin content of 1 kg food is given below

Food	Vitamin A	Vitamin B	Vitamin C
x	1	2	3
y	2	2	1

\[1\text{ }kg\]of food X costs \[Rs.\text{ }16\]and \[1\text{ }kg\] of food Y costs \[Rs.\text{ }20\]. Find the least cost of the mixture which will produce the required diet.

A)  \[Rs.\,100\]

B)  \[Rs.\,98\]

C)  \[Rs.\,116\]

D)  \[Rs.\,112\]

Correct Answer: D

Solution :

From the given data, we get the following LPP Min  \[Z=16x+20y\] S.t. \[x+2y\ge 10,\] \[2x+2y\ge 12\] \[3x+y\ge 8\] and \[x\ge 0,\,\,y\ge 0\] First we assume all the inequalities as equations                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                

Equations	Points
\[x+2y=0\]	\[(0,\,5)\] and \[(10,\,0)\]
s\[2x+2y=12\]	\[(0,6)\] and \[(6,0)\]
\[3x+y=8\]	\[(0,8)\] and \[\left( \frac{8}{3},0 \right)\]

Now, we plot all these points on graph paper for getting a convex region. For intersection point P, \[\begin{align}   & x+2y=10 \\  & 6x+12y=16 \\  & -\,\,\,\,\,\,\,\,\,\,-\,\,\,\,\,\,\,\,\,\,\,- \\  & \_\_\_\_\_\_\_\_\_\_\_ \\  & -5x=-6 \\ \end{align}\] \[\Rightarrow \] \[x=\frac{6}{5}\] and they \[y=\frac{22}{5}\] \[\therefore \]  Convex region is TSQD with extreme point \[T(0,8),\,S(1,5),\,Q(2,\,4)\] and \[D(10,0).\] Now, apply coner point method

Points	Objective function \[\min \,\,z=16x+20y\]
\[T(0,\,8)\]	\[16\times 0+20\times 8=160\]
\[S(1,\,5)\]	\[16\times 1+20\times 5=116\]
\[Q(2,\,4)\]	\[16\times 2+20\times 4=112\](min)
\[D(10,\,0)\]	\[16\times 10+20\times 0=160\]

So, the least cost of the mixture is Rs. 112.