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 Rs.17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of cach should be produced each day as to maximise his profit, if he operates his machines for at the most 12 hours a day?

Let the manufacturer produce x nuts and y bolts.
Let Z be the profit.
Table:

We are to maximize P = 
subject to constraints
x + 3 y ≤ 12
3 x + y ≤ 12
x, y ≥ 0

Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0, y ≥ 0 lies in the first quadrant.
Now we draw the graph of x + 3y = 12
For x = 0, 3 y = 12 or y = 4
For y = 0, x = 12
∴  line meets OX in A(12, 0) and OY in L(0, 4)
Again we draw the graph of
3x + y = 12
For x = 0, y = 12
For y = 0, 3x = 12 or x = 4
∴ line meets OX in B(4, 0) and OY in M(0, 12).
Since feasible region satisfies all the constraints.
∴ OBCL is the feasible region.
The comer points are O(0, 0), B(4, 0), C(3, 3), L(0, 4)


∴ maximum value of P is 73.5 at (3, 3)
∴  3 packages of nuts and 3 packages of bolts are produced for maximum profit of Rs. 73.50.