Question

A factory makes tennis rackets and cricket bats. A tennis racket takes $1.5$ hours of machine time and $3$ hours of craftmans time in 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.

Answer 1

Let number of tennis racket be made is $X$ and number of cricket bat be made is $Y$

Since, tennis bat requires $1.5$ hours and cricket bat requires $3$ hours of machine time. Also, there is maximum $42$ hours of machine time available.

$\therefore 1.5X+3Y\le 42$

$\Rightarrow X+2Y\le 28...\left(1\right)$

Since, tennis bat requires $3$ hours and cricket bat requires $1$ hours of craftmans time. Also, there is maximum $24$ hours of craftmans time available.

$\therefore 3X+Y\le 24...\left(2\right)$

Since, count of an object can't be negative.

$\therefore X\ge 0,Y\ge 0...\left(3\right)$

We have to maximize profit of the factory.

Here, profit on tennis racket is $20Rs$ and on cricket bat is $10Rs$

So, objective function is $Z=20X+10Y$

Plotting all the constraints given by equation (1), (2) and (3), we got the feasible region as shown in the image.

Corner points	Value of $Z=20X+10Y$
A $(0,14)$	$140$
B $(4,12)$	$200$ (maximum)
C $(8,0)$	$160$

Hence,

(i) $4$ tennis rackets and $12$ cricket bats must be made so that factory will work at full capacity.

(ii) Maximum profit of factory will be $200Rs$