$2)(10$ marks$)$ Martha Manufacturing makes three products. Each product requires manufacturing operations in three departments: A$,$ B$,$ and C$.$ The labor$-$hour requirements, by department, are as follows:

Department Product $1$ Product $2$ Product $3$

A $3\mathrm{.}252\mathrm{.}53\mathrm{.}20$

B $2\mathrm{.}752\mathrm{.}551\mathrm{.}10$

C $0\mathrm{.}450\mathrm{.}751\mathrm{.}50$

Max production $7010080$

During the next production period, the labor$-$hours available are $550$ in department A$,450$ in department B$,$ and $110$ in department C$.$ The profit contributions per unit are $$25$ for product$1,$ $$28$ for product $2,$ and $$30$ for product $3.$ Setup costs are $$400$ for product $1,$ $$550$ for product $2,$ and $$600$ for product $3.$

a$)$ Formulate a linear programming model for maximizing total profit contribution. $(4$ marks$)$

b$)$ Solve the linear program formulated in part $($a$).$ How much of each product should be produced, and what is the projected total profit contribution? $(6$ marks$)$