$($c$)$ Suppose that $8,6,$ 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?

If required, round your answer to two decimal places.

Product $1$ Product $2$

Production

If required, round your answer to nearest whole number.

total'Profit $

How much overtime do you recommend using in each department?

If required, round your answer to two decimal places. If you answer is zero, enter $"0".$

What is the increase in the total contribution to profit if overtime is used?

If required, round your answer to nearest whole number.