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posted by Ak on .
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?

Assistance needed.
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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(2103n)
profits = (2103n)(45+2n)  5(2103n)
= ...
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.