Suppose 70 customers are tourists and 30 are business travelers. Each type of consumer has a different willingness to pay for each class of ticket. Tourist travelers would be willing to pay up to $140 for an Economy ticket and up to $175 for a First Class ticket. Business travelers would be willing to pay up to $225 for an Economy ticket and $300 for a First Class ticket. The marginal cost to the airline of an Economy seat is $100 and $150 for a First Class seat.

Derive the optimal First Class ticket price (PFC) and the Economy ticket price (PE) that the airline should charge. Assume that the airline cannot observe which type of consumer is buying the ticket. Enumerating all possible prices that the airline could charge for each type of ticket is not an acceptable answer. Please explain your mathematical derivation