$1.$ A company manufactures two products X and Y whose profit contributions are $$10$ and $$20$ respectively. Product X requires $6$ hours on machine I, $9$ hours on machine II and $9$ hours on machine III. The requirement of product Y is $3$ hours on machine I, $6$ hours on machine II and $5$ hours on machine III. The available capacities for planning period for machines I, II and III are $30,36$ and $20$ hours respectively.

Find the optimal product mix using a graphical method.