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The Green Mountain Inn can rent all its 210 rooms when it charges $45 per night for a room, but the manager wants to increase profits. He finds, however, that for each $2 increase in the room rate, 3 fewer rooms are rented. If the cost of cleaning an occupied room is $5 per night, what should the manager charge per night for a room to maximize profits?

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    Let the number of $2 increases be n
    Now:
    number of rooms = 210
    cost per room = 45
    cost of cleaning= 210(5)
    After increase:
    number of rooms = 210 - 3n
    cost per room = 45 + 2n
    cost of cleaning = 5(210-3n)

    profits = (210-3n)(45+2n) - 5(210-3n)
    = ...
    expand the above, then express as a simplified quadratic.
    If you know Calculus, differentiate and set equal to zero to solve for n
    If you are doing these by completing the square, do so.

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