A Delta flight from ATL to PUJ $($Punta Cana, Dominican Republic$)$ has a capacity of $100$ 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 $$750.$ In the past, no$-$shows have ranged uniformly from $20$ to $30.$ If Delta cannot accommodate a passenger on this flight, the passenger will be sent to Miami on a Delta flight and then to San Juan, PR on an American Airlines flight. The passenger will spend the night at a San Juan airport hotel. The next morning, the passenger will be flown by American to Punta Cana. In addition to the denied boarding compensation $($DBC$)$ based on the US$-$DOT formula, Delta airline needs to pay American Airlines $$800$ for re$-$accommodation. The hotel and food vouchers cost $$150.$ 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$?$