A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman's 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, then find the maximum profit of the factory when it works at full capacity.

.: Let the factory make ‘x’ tennis rackets and ‘y’ cricket bats.

Clearly $x \ge 0,y \ge 0.$ We make the following table from the given data

Since the availability in the factory is not more than

42 hours of machine time and 24 hours of craftsman's time, we have the constraints :

$1.5x + 3y \le 42$

$3x + 7y \le 24$

(i) Total number of bats and rackets made in a factory is $Z = x + y$

Hence, the mathematical formulation of the problem is :

Maximize $Z = x + y$

...(1)
subject to the constraints
$x + 2y \le 28$ ...(2)

$3x + y24$ ...(3)

$x,y \ge 0$ ...(4)

Let ${l_1}:x + 2y = 28,$

${l_2}:3x + y = 24$
Let us graph the inequalities (2) to (4).

The feasible region determined by the system is shown in the graph.

Here, observe that the feasible region is bounded.

Hence, ${Z_{\max }} = 16$

i.e. 4 tennis rackets and 12 cricket bats must be made so that the factory works at full capacity.

(ii) Profit function, $Z = 20x + 10y$

Applying Corner Point Method,

we have

Hence, ${Z_{\max }} = Rs.200$

when 4 tennis rackets and 12 cricket bats are made.