# In order to supplement daily diet,a person wishes to take some X and some wishes Y tablets. The person needs atleast 18 milligrams of iron ,21 milligrams of calcium and 16 milligram of vitamins.The price of each tablet of X and Y is Rs.2 and Rs.1 respectively.How many tablets of each should the person take in order to satisfy the above requirement at the minimum cost? The contents of iron ,calcium and vitamins in X and Y(in milligrams per tablet) are given as below:  $\begin{matrix} \underline{\text{Tablets} }& \underline{\text{Iron}}&& \underline{\text{Calcium}} && \underline{\text{Vitamin}} \\ \text{X}& 6 &&3 && 2 \\ \text{Y}& 2 &&3 && 4 \\ \end{matrix}$

Toolbox:
• Let $R$ be the feasible region for a linear programming problem and let $z=ax+by$ be the objective function.When $z$ has an optimum value (maximum or minimum),where variables $x$ and $y$ are subject to constraints described by linear inequalities,this optimum value must occur at a corner point of the feasible region.
• If R is bounded then the objective function Z has both a maximum and minimum value on R and each of these occur at corner points of R
Step 1:
A person needs at least 18mg of iron.
Hence $6x+2y\geq 18$
The person needs at least 21mg of calcium.
Hence $3x+3y\geq 21$
The person needs at least 16mg of vitamins .
Hence $2x+4y\geq 16$
The price of each tablet of $X$ is Rs.2 and tablet $Y$ is Rs.1
Hence $Z=2x+y$
Step 2:
Let us draw the graph for the lines AB :$6x+2y=18,CD :3x+3y=21$ and $2x+4y=16$
Consider the line $AB :6x+2y=18$
Put $x=0,y=0\Rightarrow 0\geq 18$
Which is not true.
Hence the region $6x+2y\geq 18$ lies above the line $6x+2y=18$
Consider the line $CD :3x+3y=21$
Put $x=0,y=0\Rightarrow 0\geq 21$
Which is not true.
Hence the region $3x+3y\geq 21$ lies above the line $3x+3y=21$
Consider the line $EF :2x+4y=16$
Put $x=0,y=0\Rightarrow 0\geq 16$
Which is not true.
Hence the region $2x+4y\geq 16$ lies above the line $2x+4y=16$
Clearly the feasible region s the shaded region which is not include the region.
The corner points of the feasible region is $A(0,9),P(1,6),Q(6,1),F(8,0)$
Step 3:
Let us obtain the value of objective function $Z=2x+y$ as follows :
At the points (x,y) the value of the objective function subjected to $Z=2x+y$
At $A(0,9)$ the value of objective function $Z=2\times 0+9=9$
At $P(1,6)$ the value of objective function $Z=2\times 1+6=8$
At $Q(6,1)$ the value of objective function $Z=2\times 6+1=13$
At $F(8,0)$ the value of objective function $Z=2\times 8+0=16$
Clearly the person should take one tablet of $X$ and 6 tablet of $Y$ to satisfy the requirement at the minimum cost.