# This problem is driving me CRAZY!(CALCULAS)

Ok, so I've looked through all of my notes and throughout the text book and I cannot find how to do this problem. I'm sure it's probably pretty easy, but I just can't even figure out how to start. Here it is:

Gritz-Charlston is a 300 unit luxury hotel. All rooms are occupied when the hotel charges $80 per day for a room. For every increase of x dollars in the daily room rate, there are x rooms vacant. Each occupied room costs$22 per day to service and maintain. What should the hotel charge per day in order to maximize profit?

If anyone could just get me started it would be greatly appreciated.

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First, write down the profit function P. The key word is 'maximize' which means you should take the derivative of P and set it equal to zero (slope = 0) to find the price the hotel should charge.

So P is a function of price.

Lol, no. That's your job! Let me see what you come up with, and we'll go from there.

done

rooms: 300
vacant rooms = increment of price: x
occupied rooms: y = 300-x
constraint: x<=300

profit per room: (80+x)*y = (80+x)*(300-x)
expenses: 22*y = 22*(300-x)

total profit: per room + expenses
total profit: (80+x)*(300-x) - 22*(300-x)
total profit: 24000 + 220x - x^2 - 6600 + 22x
total profit: - x^2 + 242x + 17400

optimize that one - differentiate and solve
diff: -2x + 242 + 0
242 - 2x= 0 =>
x = 121

Profit is 32041 per day/night.