The Mountain Top Hotel is a luxury hotel in a popular ski resort area. The hotel always is essentially full during winter months,so reservations and payments must be made months in advance for week$-$long stays from Saturday to Saturday. Reservations can be canceled until a month in advance but are nonrefundable after that. The hotel has $100$ rooms and the room charge for a week's stay is $$\backslash$$ $3,000$$$.$ Despite this high cost, the hotel's wealthy customers occasionally will forfeit this money and not show up because their plans have changed. On the average, about $10$ percent of the customers with reservations are no$-$shows, so the hotel's management wants to do some overbooking. However, it also feels that this should be done cautiously because the consequences of turning away a customer with a reservation would be severe. These consequences include the cost of quickly arranging for alternative housing in an inferior hotel, providing a voucher for a future stay, and the intangible cost of a massive loss of goodwill on the part of the furious customer who is turned away $($and surely will tell many wealthy friends about this shabby treatment$).$ Management estimates that the cost that should be imputed to these consequences is $$\backslash$$ $20,000$$$.$

Use the overbooking model presented in Sec. $18\mathrm{.}8,$ including the normal approximation for the binomial distribution, to determine how much overbooking the hotel should do$.$