What is underbooking in hotel

Hotel manager Mr. Smith and his resourceful assistant, John, run a 26-room hotel in a little town. A combination of Mr. Smith's friendly attitude and the absence of a respectable hotel in the nearby vicinity imply that Mr. Smith enjoys sufficient demand at his low fare of $159 per night. John notes that some customers will walk into the hotel requesting a room for that evening and they are willing to pay a high fare of $325 per night. John knows this demand is variable. (In reality, this demand is U(4,9)) He suggests some rooms should be kept unsold to the low-fare customers so that they can serve the high-fare customers. Please solve A, B, and C. Please show work. Thank you.

A. To maximize profits with John's plan, what is the booking limit that should be set for low fare customers?

B. Mr Smith does not like the idea of John and says “we will sell to everyone who reserves in advance and ignore the walk-in demand." If Mr Smith has his way, what will be the hotel's expected revenue?

C. John replies, "If you are going to forgo the opportunity to sell to last-minute customers, let's at least accept more than 26 reservations for the evening." Checking the data, John observes that the number of "no-shows" is uniformly distributed with U(1,8). (Recall, a "no-show" is when a customer makes a reservation but doesn't show up to use the room that evening.) John also notes that a $100 non-refundable deposit is required with all reservations. However, if the hotel does not have a room for a reservation holder, then they need to book that person in a B&B in the nearest town. They decide that in those cases they would refund the customer's deposit and pay for the customer's stay in the B&B, which is $450. The customer would not be happy, but they are getting a free night, so John figures that there would be no loss of goodwill. Finally, if they have an empty room due to a no-show, they also figure that they would not be able to fill the room with a last-minute customer. What is the overbooking quantity to maximize revenue?