Expert: Scotto - 11/12/2010

Question
A 150-room resort hotel is filled at a room rate of $125 per day. For each $5 increase in the room rate, three fewer rooms are rented. What room rate will result in maximum daily revenue? How many rooms will be rented at that rate?

Answer
Since the increase is $5 and the decrease is 3 rooms, let n be the variable.  This gives the rental rate as 125 + 5n and the number of rooms at 150 - 3n.

To get the amount of money made, multiply these together.
This gives (125+5n)(150-3n) = 18,750 + 750n - 375n - 15n^2.
Combining the n's gives 18,750 + 375n - 15n^2.

Since there is a negative on the n^2, it is a downward facing parabola.  Taking the derivative and setting it to 0 will give the x value that the maximum occurs at.

Let f(n) = 18,750 + 375n - 15n^2, then f'(n) = 375 - 30n.
Setting this to 0 gives 30n = 375, or n = 12.5.
This would give the number of rooms as 37.5.

Since n is related to the number of rooms, try n = 12 and n = 13 to see which gives a greater value.  Going from n = 11 to n = 14 gives
  11   $21,060,
  12   $21,090,
  13   $21,090, and
  14   $21,060.

As can be seen, n needs to be 12 or 13.


If the problem is looked at in more detail, n could possibly be a real number since an increase or decrease of 1/3 would give an integer number of rooms.  This would mean that n = 12 1/3 or 12 2/3.
It can now be seen that each of these gives %21,093.33.