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A 300 room hotel is filled to capacity at \$80 a night. If the charge is increased by \$3 it rents 9 less rooms. If it costs \10 to clean a rented room the next day, how much should the inn keeper charge in order to maximize its profit?

I thought the question was really straight forward and that I'd be able to do the following to get my answer:

Revenue = (# Of Rooms * Room Charge) - (# Of Rooms * Clean Charge)

Full Inn

Revenue = (300 * 80) - (300 * 10) which is 24,000 - 3,000 so Revenue = \21,000

Not Full Inn

Revenue = (291 * 83) - (291 * 10) which is 24,153 - 2,910 so Revenue = \21,243

Therefore the inn keeper should charge \83 a room. I got the question wrong, so can someone explain what I should have done?

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    Instead of my $x$, you may wish to replace it everywhere by $3x$. So $80+3x$, rent $300-9x$, and so on. I used my version because it makes for marginally simpler-looking numbers. Kind of an automatic reflex!2012-12-05

1 Answers 1

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Let us charge $80+x$. Then we rent $300-3x$ rooms.

Net Income, after cleaning: $(80+x)(300-3x) -(10)(300-3x)$.

Do the usual stuff to maximize, not forgetting about endpoints. There is also the complication that the number that maximizes our function will not necessarily lead to an integer number of rooms rented, so we may have to make a mild adjustment.

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    In the problem, we are told $3$ dollars more, occupancy goes down by $9$. So $1$ dollar more, occupancy goes down by $3$. So $x$ (my $x$) dollars more, occupancy is $300-3x$. Makes calculation easier.2012-12-05