$[$Q $6-10]$ A newly created NorthEast Airways $($NE$)$ flight from Philadelphia to Boston has $300$ seats. The high fare on the flight is $$800$ and the restricted low fare is $$300.$ There is ample demand for the low fare class but high fare demand is random. Furthermore, the customers who buy low fare buy their tickets well in advance before high fare customers. Assume the demand for the high fare is normally distributed with a mean $120$ and a standard deviation of $50.$

Due to the problem of passenger no$-$shows, Mr$.$ Wright decides to overbook the flight. He estimates that his loss from a passenger holding a valid reservation who is denied a seat is $$550.$ He also estimates that the number of passengers with reservations who are no$-$shows is Poisson with a mean of $9\mathrm{.}25.$ What is Mr$.$ Wright$$s optimal overbooking quantity?