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 Rsl7.50 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?

Explanation

Let number of packages of nuts produced be `x`
and number of packages of bolts produced be `y`



According to Question:

Machine A
Time to produce nuts `to` 1 hrs
Time to produce bolts`to` 3 hrs
Maximum Time Available `to` 12 hrs
`therefore x+3y<=12`

Machine B
Time to produce nuts `to` 3 hrs
Time to produce bolts`to` 1 hrs
Maximum Time Available `to` 12 hrs
`therefore 3x+y<=12`

Also, `x>=0,y>=0`

As, we need to Maximize the profit, the function used here will be Maximize `Z`

Profit on each Nut `to` Rs 17.50
Profit on each bolt `to` Rs 7.00

`therefore` Maximize `Z=17.5x+7y`

Combining all constraints:
Maximize `Z=17.5x+7y`
`x+3y<=12`
`3x+y<=12`
`x>=0,y>=0`



Hence, the profit will be maximum if the company produces:
` " " " " ` Number of bolts `to` 3 packages
` " " " " ` Number of nuts `to` 3 packages
` " " " " ` Maximum Profit = Rs 73.50