William J Hill runs a small batch artisanal bourbon distillery at a secluded location in the hills of Kentucky. He makes two products, known among his customers as Rotgut and White Lightning. The recipes for the two have been passed down in the Hill familyfor generations and are Rotgut: $1$ bushel of corn, $3$ pounds of sugar, $2$ hours of cooking time. For the premium blend, White Lightning, he needs $2$ bushels of corn, $2$ pounds of sugar, and $3$ hours of cooking time. Both recipes make enough to fill two jugs, which sell for $$8$ apiece for Rotgut and $$12$ apiece for White Lightning. A quick inventory one crisp autumn morning reveals that William has on hand $40$ bushels of corn, $70$ pounds of sugar, and $50$ jugs. He would like to brew up a few artisanal batches, but has recently received a tip that certain agencies have taken an interest in his talents and may be paying him a visit in three days, hence he plans to restrict any brewing activity to $72$ hours at most, before he retreats to his home away from home, Lubbock. William cleans his equipment, lights a fire, and ponders the objectives. Obviously the first priority is restricting himself to $72$ hours of work $$any more than that and he runs the risk of an extended holiday. His second priority to make enough to acquirematerials for the next production run and fund his daughter's college tuition $-$he believes that $$500$ would make this production run worth his while. His third and fourth priorities are not to have too much perishable inventory, so he wants to make sure he doesn't have too many bushels of corn on hand $($third priority$)$ nor does he want too many pounds of sugar on hand $($fourth priority$).$

$\mathrm{.}3\mathrm{.}1$ Which is the appropriate objective function for this scenario?

$3\mathrm{.}2$ Which is the appropriate profit constraint for this scenario?

$3\mathrm{.}3$ Which is the appropriate time constraint for this scenario?

$3\mathrm{.}4$ Which is the appropriate corn constraint for this scenario?