Question

A factory makes cricket bats and hockey sticks. A bat takes $1.5$ hours of machine time and $2$ hours of craftsman time. While a hockey stick takes $2.5$ hour of a machine time and $1.5$ hours of craftsman time. In a day the factory is available up to $80$ hours of machine time and $70$ hours of craftsman time. Show that in order to make maximum profit the factory should produce only cricket bats given that the profits on a cricket at and a hockey stick are Rs.$50$ and Rs. $35$ respectively.

Answer 1

Let the number of cricket bat be $x$ and hockey sticks be $y$

In the factory machine time is available upto $80$ hours

$1.5x+2.5y\le 80$

In the factory craftsman time is available upto $70$ hours

$2.6x+1.5y\le 70$

Let the total profit be $z$.

Profit on cricket bat is $Rs.50$ and $Rs.35$ on hockey stick.

$z=50x+35y$

Maximize$z=50x+35y$

Subject to $1.5x+2.5y\le 80$

$2x+1.5y\le 70$

$1.5x+2.5y\le 80$

$x$	$0$	$20$
$y$	$32$	$20$

$2x+1.5y\le 70$

$x$	$35$	$20$
$y$	$0$	$20$

From the graph we can conclude the maximum profit is at $(20,20)$$

So, $20$ cricket bat and $20$ hockey sticks should be produced for maximum profit.