Step 1:

The given problem can be written as follows :

Type A screws :Threading machine requires 2 min,slotting machine requires 3 min,cost per box =Rs.100

Type B screws :Threading machine requires 8 min,slotting machine requires 2 min,cost per box =Rs.170

The mathematical formulation for the above problem is as follows :

Let $x$ be the no of screws of type A

Let $y$ be the no of screws of type B

Time taken for the screws of type A and type B to be allowed in threading machine is 2min and 8 min respectively.

Time taken for the screws of type A and type B to be allowed in slotting machine is 3min and 2 min respectively.

The time available in one week is 60 hour (i.e.)3600min

The cost per box for type A screws is Rs 100 and type B box is Rs.170

We have the maximize the profit

Hence the objective function is $Z=100x+170y$ subjected to constraints.

$2x+8y\leq 3600,3x+2y\leq 3600$ and $x,y\geq 0$

Step 2:

Let us draw the graph for the lines $AB :2x+8y=3600$ and $CD :3x+2y=3600$

Consider the line $AB :2x+8y=3600$

Put $x=0,y=0$ then $0\leq 3600$ is true.

Clearly the region $2x+8y\leq 3600$ lies below the line AB.

Consider the line $CD :3x+2y=3600$

Put $x=0,y=0$ then $0\leq 3600$ is true.

Hence the region $3x+2y\leq 3600$ lies below the line CD.

The origin O(0,0) is also included in the feasible region.

The feasible region OCPB is the shaded portion shown in the fig.

The point of intersection of the two lines is $P(1080,180)$

Hence the corner points of the feasible region OCPB are $O(0,0),C(0,450),P(1080,180),B(1200,0)$

Step 3:

To find the value of the objective function $Z=100x+170y$ as follows :

At the points $(x,y)$ the value of the objective function subjected to $Z=100x+170y$

At $O(0,0)$ the value of the objective function $Z=0$

At $C(0,450)$ the value of the objective function $Z=100(0)+170(450)=76500$

At $P(1080,180)$ the value of the objective function $Z=100(1080)+170(180)=138600$

At $B(1200,0)$ the value of the objective function $Z=100(1200)+0=120000$

Hence the maximum value of $Z$ is at $P(1080,180)$ .

The maximum profit is Rs.138600$