A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftmans time m its making while a cricket bat takes 3 hour of machine time and 1 hour of craftmans time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsmans time.(i)         What number of rackets and bats must be made if the factory is to work at full capacity?(ii)        If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.

Explanation

Let Number of Tennis Racket be x, and
` " " `Number of Cricket Bats be y



According to question:
Machine Hours
Tennis Racket requires `to` 1.5 hrs
Cricket Bat requires `to` 3 hrs
Maximum time available `to`42 hrs
`therefore`
`1.5x+3y<=42`
`3x+6y<=84`
`x+2y<=28`

Craftsman's Hours
Tennis Racket requires `to` 3 hrs
Cricket Bat requires `to` 1 hrs
Maximum time available `to`24 hrs
`therefore`
`3x+y<=24`

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

As we want to maximize the profit.
Hence, the function used here will be Maximize Z

Profit on each tennis racket `to` Rs 20
Profit on each cricket Bat `to` Rs 10
`therefore`
Maximize `Z=20x+10y`
Subject to constraints,
`x+2y<=28`
`3x+y<=24`
`x>=0, y>=0`







Thus,

(i) 4 Tennis Rackets and 12 Cricket Bats must be made so that factory runs at full capacity.

(ii) Maximum Profit is Rs 200, when 4 tennis rackets and 2 cricket bats are produced.