Suppose that $10,6,$ and $8$ hours of overtime may be scheduled in departments A$,$ B$,$ and C$,$ respectively. The cost per hour of overtime is $$18$ in department A$,$ $$22\mathrm{.}50$ in department B$,$ and $$12$ in department C$.$ Formulate a linear programming model that can be used to determine the optimal production quantities if overtime is made available. What are the optimal production quantities, and what is the revised total contribution to profit?