Question

A dietician has to develop a special diet using two foods $P$ and $Q$. Each packet (containing $30g)$ of food $P$ contains $12$ units of calcium, $4$ units of iron, $6$ units of cholesterol and $6$ units of vitamin $A$. Each packet of the same quantity of food $Q$ contains $3$ units of calcium, $20$ units of iron, $4$ units of cholesterol and $3$ units of vitamin $A$. The diet requires at least $240$ units of calcium, at least $460$ units of iron and at most $300$ units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin $A$ in the diet? What is the minimum amount of vitamin $A$?

Answer 1

Simplification of given data:
Let the number of packets of food $P$ be $x$ and number of packets of food $Q$ be $y$


Combining all the constraints:
Minimize $Z=6x+3y$
Subject to constraints,
$12x+3y\ge 240$
$\Rightarrow 4x+y\ge 80,$
$4x+20y\ge 460$
$\Rightarrow x+5y\ge 115,$
$6x+4y\le 300$
$3x+2y\le 150,$
$x\ge 0,y\ge 0$

Draw graph
$4x+y=80$
$\begin{array}{ccc}x& 0& 20\\ y& 80& 0\end{array}$

$x+5y=115$
$\begin{array}{ccc}x& 0& 70\\ y& 23& 9\end{array}$

$3x+2y=150$
$\begin{array}{ccc}x& 0& 50\\ y& 75& 0\end{array}$


Find minimum amount of Vitamin $A$

$\begin{array}{cc}\text{Corner Points}& \text{Value of}Z=6x+3y\\ (2,72)& 228\\ (15,20)& 150\\ (40,15)& 285\end{array}$

Hence, the amount of Vitamin $A$ will be minimum if
$15$ packets of Food $P$ $\&$ $20$ packets of Food $Q$ are used. Thus, Minimum amount of Vitamin $A=150$ units

Therefore, Minimun amount of Vitamin
$A=150$ units