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 per package of bolts. How many packages of each should be produced each day so as to maximize his profits if he operate his machines for at the most 12 hours a day?

3

Let x package nuts and y package bolts are produced

Let z be the profit function, which we have to maximize.

Here, $z=17.50x+7y$....(1) is objective function.

And constraints are

$x+3y≤12$ .... (2)

$3x+y≤12$ .... (3)

$x≥0$ ....(4)

$y≥0$ ....(5)

On plotting graph of above constraints or inequalities (2),(3),(4) and (5) we get shaded region as feasible region having corner points A, O, B and C.

For coordinate of 'c' two equations,

$x+3y=12$....(6)

$3x+y=12$....(7)

On solving we get, $x=3$ and $y=3$

Hence coordinate of C are (3,3)

Now value of z is evaluated at corner point as shown in the graph.

Therefore, maximum profit is Rs. 73.5 when 3 package nuts and 3 package bolt are produced.