Question

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 $1kg$ food is given below

• A. $Rs.100$
• B. $Rs.98$
• C. $Rs.116$
• D. $Rs.112$

Answer 1

Answer: (D)

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, $\begin{array}{rl}& x+2y=10\\ & 6x+12y=16\\ & ---\\ & \_\_\_\_\_\_\_\_\_\_\_\\ & -5x=-6\end{array}$ $\Rightarrow$ $x=\frac{6}{5}$ 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.