A 300 room hotel in Macau is filled to full capacity every night at $800 per room. For each additional $10 increase in rent, 3 fewer rooms are rented. If each rented room costs $160 to service per day, how much should the hotel charge for each room to maximize profit?

Question

anonymous · Answer

P = (300 - 3x)*(800 -160 + x)

anonymous · Answer

(300 - 3x) is the number of rooms rented
(800 - 160 + x) is the profit per room

anonymous · Answer

simplify that, factor it out, take it's derivative, and find the critical points for that derivative

anonymous · Answer

that help?

anonymous · Answer

oh and every "x" represents a 10$ increase per room

anonymous · Answer

hmm that means x=$10?
so i just put into the equation.and how to do next?
i dont really get the next part.

anonymous · Answer

FOIL

anonymous · Answer

your learning derivatives, right? After you get it into a polynomial, the derivative is pretty easy

anonymous · Answer

hmm  thats mean expand the (300-3x)(800-160x)
and differentiate the expansion.
and equate its to 0 right?
btw how do you get the (300-3x)(800-160x)?I dont get this step.

anonymous · Answer

Sorry I was a little off. its$$P=(300 - 3x)(800 - 160 + 10x)$$which, by subtracting 160 from 800, is equal to $$P=(300 - 3x)(640 + 10x)$$which, by F.O.I.L. is equal to $$P=(300*640) + 3000x -(640*3*x) - 30x^2$$or$$P=-30x^2 + 1080x +192000$$then take the derivative, which is $$\frac{dP}{dx}=-60x+1060$$Then find the critical points by seeing where this equals zero.  Understand that the derivative of the profit function shows the rate that profit is changing with respect to x.  So when the derivative reaches zero, the hotel is no longer profiting by charging more per night.$$0=-60x+1060$$$$x=17\frac{2}{3}$$Round that to 18.  Therefore, the hotel maximizes profit with eighteen 10$ increments above 800$ per night, or at 980$ per night.

anonymous · Answer

EDIT: I said 1060 twice where it should have been 1080.  The answer is still the same.

anonymous · Answer

Thanks!!

phi · Answer

Maybe we can tweak this answer?
Let x = the price increase over $800
Then the profit from each rented room is the amount brought in minus the cost of service.  amount brought in is 800+x , less 160. so
(800-160+x)= (640+x)
The number of rooms rented is reduced by 3 for every increase by $10
That is, if x=10, we expect 3*x/10 fewer rooms i.e. 3*10/10 = 3 fewer rooms
so the equation we have is
profit= (300-3x/10)(x+640)

Does this sound right?

anonymous · Answer

Is that an improvement, or just a different definition of x?  Our equations are identical except I have defined x to be 10$ increments, so as to not have to say 3x/10

phi · Answer

Your right!  I saw the 17 2/3 (which I now see came from a typo), and jumped to the conclusion the equation was wrong.  I apologize for stepping on your toes, metaphorically speaking...