P$4(20$ points$)$: The following diagram shows the flow process, raw material costs, and machine processing time for three products: $\backslash (\backslash$mathrm$\{$A$\},\backslash$mathrm$\{$B$\}\backslash ).$ and C $.$ There are three machines $(\backslash (\backslash$mathrm$\{$W$\},\backslash$mathrm$\{$X$\}\backslash ),$ and Y $)$ used in the production of these products: the times shown are in required minutes of production per unit. Raw material costs are shown in cost per unit of product. The market will take all that can be produced.

a$)$ On the basis of maximizing gross profit per unit, which product should be sold?

b$)$ To maximize total profit for the firm, which product should be sold?