# 4.    A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours 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 Rs17.50 per package on nuts and Rs 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machines for at the most 12 hours a day?

S seema garhwal

Let packages of nuts be x and packages of bolts be y  .Thus, $x\geq 0,y\geq 0$.

The given information can be represented in table as :

 bolts nuts availability machine A 1 3 12 machine B 3 1 12

Profit on a package of nuts is Rs. 17.5  and on package of bolt is 7.

Therefore, constraint  are

$x+3y\leq 12$

$3x+y\leq 12$

$x\geq 0,y\geq 0$

$Z= 17.5x+7y$

The feasible  region determined by constraints is as follows:

The corner points of feasible region are  $A(4,0),B(3,3),C(0,4),D(0,0)$

The value of Z at corner points is as shown :

 Corner points $Z= 17.5x+7y$ $A(4,0)$ 70 $B(3,3)$ 73.5 maximum $C(0,4)$ 28 $D(0,0)$ 0

The maximum value of z is 73.5 at $B(3,3)$.

Thus, 3 packages of nuts and 3 packages of bolts should be manufactured everyday to get maximum profit.

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