Question

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, while each packet of the same quality of food $Q$ contains $3$ units of calcium, $20$ units of vitamin A. The diet requires atleast $240$ units of calcium, atleast $460$ units of iron and almost $300$ units of cholesterol. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A?

Answer 1

Let $x$ and $y$ be the number of packets of food $P$ and $Q$ respectively.

Obviously $x\ge 0,y\ge 0$.

Mathematical formulation of the given problem is as follows:
Minimise $Z=6x+3y$ (vitamin A)

subject to the constraints
$12x+3y\ge 240$ (constraint on calcium), i.e. $4x+y\ge 80$ ... (1)
$4x+20y\ge 460$ (constraint on iron), i.e. $x+5y\ge 115$ ... (2)
$6x+4y\le 300$ (constraint on cholesterol), i.e. $3x+2y\le 150$ ... (3)
$x\ge 0,y\ge 0$ ... (4)

$Z$ is minimum at the point $(15,20)$.

Hence, the amount of vitamin A under the constraints given in the problem will be minimum, if $15$ packets of food $P$ and $40$ packets of food $Q$ are used in the special diet.

The minimum amount of vitamin A will be $285$ units.