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Homework Help: This problem is driving me CRAZY!(CALCULAS)

  1. Mar 17, 2008 #1
    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.
  2. jcsd
  3. Mar 17, 2008 #2
    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.
  4. Mar 17, 2008 #3
    Thanks, could you give me the equation to start with?
  5. Mar 17, 2008 #4
    Lol, no. That's your job! Let me see what you come up with, and we'll go from there.
  6. Mar 18, 2008 #5

    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.
  7. Mar 18, 2008 #6
    Don't give out answers...
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