The owner of Genuine Subs, Inc., hopes to expand the present operation by adding one new outlet. She has studied three locations. Each would have the same labor and materials costs $($food$,$ severing containers, napkins, etc.$)$ of $$1\mathrm{.}76$ per sandwich. Sandwiches sell for $$8$ each in all locations. Rent and equipment costs would be $$5,000$ per month for location A$,$ $$5,500$ per month for location B$,$ and $$5,800$ per month for location C$.$ a$.$ Determine the volume necessary at each location to realize a monthly profit of $$10,000.$

b$.$ If expected sales at A$,$ B$,$ and C are $21,000$ per month, $22,000$ per month, and $23,000$ per month, respectively, which location would yield the greatest profits?