$($a$)$ A cruise line offers a trip for $$2000$ per passenger. If at least $100$ passengers sign up$,$ the price is reduced for all the passengers by $$10$ for every additional passenger $($beyond $100)$ who goes on the trip. The boat can accommodate $250$ passengers. What number of passengers maximizes the cruise line's total revenue? What price does each passenger pay then?

The number of passengers that maximizes the cruise line's total revenue is

The price the passengers pay is $

$($b$)$ The cost to the cruise line for $n$ passengers is $80,000+500n.$ What is the maximum profit that the cruise line can make on one trip? How many passengers must sign up for the maximum to be reached and what price will each pay?

The maximum profit that the cruise line can make on one trip is $

The number of passengers that must sign up for the maximum to be reached is $,$ and each passenger mi

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