Q$1.(50$ points$)$ A small firm builds two types of products. Type A requires $20$ minutes on threading machine, $10$ minutes on slotting machine and $7\mathrm{.}5$ minutes of craftsman time. Type B requires $10$ minutes on threading machine, $20$ minutes on slotting machine and $30$ minutes of craftsman time.

Each day there are $100$ minutes of threading machine time, $150$ minutes of slotting machine time and $180$ minutes of craftsman time available. The profit on each type A product is $$5$ and on each type B product is $$12.$

To maximize profit, please:

a$.$ Formulate the appropriate linear programming problem.

b$.$ Use graphical solution to solve it$,$ please mark feasible set and corner points clearly.

Q$2.(50$ points$)$ A diet is to contain at least $150$ units of vitamins, $180$ units of minerals, and $120$ calories. Two foods, Food A and Food B are to be purchased. Each unit of Food A provides $50$ units of vitamins, $90$ units of minerals, and $30$ calories. Each unit of Food B provides $20$ units of vitamins, $20$ units of minerals, and $40$ calories. Food A costs $$2$ per unit and Food B cost $$1$ per unit. How many units of each food should be purchased to keep costs at a minimum?

a$.$ Formulate the appropriate linear programming problem.

b$.$ Use graphical solution to solve it$,$ please mark feasible set and corner points clearly.