A Delta flight from ATL to PUJ $($Punta Cana, Dominican Republic$)$ has a capacity of $200$ passengers and fills up quickly for the Thanksgiving holiday. Delta does not want any seats to be wasted on this flight due to no$-$shows. The ticket costs $$350.$ In the past, no$-$shows have followed a normal distribution with a mean of $10$ and a standard deviation of $4.$ If Delta cannot accommodate a passenger on this flight, the passenger will be sent to Miami on a Delta flight and then to San Juan, PR on an American Airlines flight. The passenger will spend the night at a San Juan airport hotel. The next morning, the passenger will be flown by American to Punta Cana. In addition to the denied boarding compensation $($DBC$)$ based on the US$-$DOT formula, Delta airline needs to pay American Airlines $$500$ for re$-$accommodation. The hotel and food vouchers cost $$100.$ Compute the optimal overbooking limit$.$

$($a$)$ What is the DBC based on the US$-$DOT formula?

$($b$)$ What is the total denial cost?

$($c$)$ What is the optimal overbooking limit$?$