Question

A sweet-shop makes gift packet of sweets by combining two special types of sweets $A$ and $B$ which weigh $7kg$. Atleast $3kg$ of $A$ and no more than $5kg$ of $B$ should be used. The shop makes a profit of Rs. $15$ on $A$ and Rs. $20$ on $B$ per $kg$. Determine the product mix so as to obtain maximum profit.

Answer 1

Solution:

Let amount of $A$ type of sweets $=xkg$

Let amount of $B$ type of sweets $=ykg$

$\therefore x\ge 0,y\ge 0$ $--------\left(I\right)$

In one gift packet both types of sweets are to be included but not more than $7kg.$

$\therefore x+y\le 7$ $------\left(II\right)$

$A$ type of sweets must be at least $3kg.$

$\therefore x\ge 3$ $------\left(III\right)$

$B$ type of sweets no more than $5kg.$

$\therefore y\le 5$ $--------\left(IV\right)$

Profit Rs. $15$ on $A$ and Rs. $20$ on $B$ per $kg$.

$\therefore$ Profit function $f=15x+20y$

The corner points for the graph are $(3,4),(3,0)$ and $(7,0)$

$x$	$y$	$f=15x+20y$
$3$	$4$	$125$
$3$	$0$	$45$
$7$	$0$	$105$

Maximum Profit of Rs. $125$ can be gained when the mixture contains $3kg$ of $A$ and $4kg$ of $B.$