Question

A manufacturer produces nuts and bolts. It takes $1$ hour of work on machine $A$ and $3$ hours of work on machine $B$ to produce a package of nuts. It takes $3$ hours on machine $A$ and $1$ hour on machine $B$ to produce a package of bolts. He earns a profit of Rs $17.50$ per package on nuts and Rs $7.00$ per package on bolts.Then number of packages that can be produced to maximize his profits, if he operates each machine for at most $12$ hours a day, is

Answer 1

Let $x,y$ packages of nuts and bolts respectively be produced.
Now, maximum working hour of both machines is $12.$
For machine $A:x+3y\le 12\cdots \left(i\right)$
For machine $B:3x+y\le 12\cdots \left(ii\right)$
and $x,y\ge 0\cdots \left(iii\right)$
Objective function $:Z=17.5x+7y$

Now, plotting the graph of given constraints :

We get feasible region as $OBEC.$
Now, value of objective function $Z=17.5x+7y$ at corner points is tabulated below.

Corner points : $(x,y)$	Value : $Z=17.5x+7y$
$O(0,0)$	$0$
$B(0,4)$	$28$
$C(4,0)$	$70$
$E(3,3)$	$73.5$

So, maximum profits can be made by producing $3$ packages of nuts and $3$ packages of bolts.
$\therefore 6$ packages should be produced daily.