A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs. 7 and screws B at a profit of Rs 1. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.

Explanation

Let the number of packages of screw A is `x`
and the number of packages of screw B is `y`



According to question

Automated Machine
Works on Screw A `to` 4 min
Works on Screw B `to` 6 min
Maximum Time `to` 240 min
`therefore 4x+6y<=240`
`Rightarrow 2x+3y<=120`
` x>=0, y>=0`

Hand Operated Machine
Works on Screw A `to` 6 min
Works on Screw B `to` 3 min
Maximum Time `to` 240 min
`therefore 6x+3y<=240`
`Rightarrow 2x+y<=80`
` x>=0, y>=0`

As, we need to maximize the profit, the function used here is Maximize `Z`

Profit on screw A `to` Rs. 7
Profit on screw B `to` Rs. 10

`therefore` Maximum `Z=7x+10y`

Combining all constraints
Max `Z=7x+10y`
`2x+3y<=120`
`2x+y<=80`
>` x>=0, y>=0`







Hence, Profit will be maximum if capacity produces:

` " " " " 30` packages of Screw `A`
` " " " " 20` packages of Screw `B`

Maximum Profit = Rs 410