A small business enterprise makes shoes and boots. To make one pair of shoes requires $\frac{1}{2}$ hour of cutting and $20$ minutes of stitching. To make a pair of boots requires $15$ minutes of cutting and $\frac{1}{2}$ hour of stitching. The profit on a pair of shoes is $80$ and on a pair of trousers $100.$ The business operates for a maximum of $8$ hours per day.

A$.$ Translate the problem into a linear programming one, identifying and writing down the objective function and the constraints.

$(5$ marks$)$

Plot a graph to identify the feasible region.

$(5$ marks$)$

Algebraically find the optimum number of pairs of shoes and boots that should be sold to maximise profits.

$(5$ marks$)$

What is the maximum profit that satisfies the objective function?

$(5$ marks$)$