Calculus optimization help?

Apr 2016
6
0
Rhode Island
I have this problem for my calculus project that I don't understand how to do, can someone help?

I have a hotel room that costs \$139.99 for one night and the hotel has 66 rooms. The problem says: The industry standard is commonly believe to be that for every \$10 increase in room rate, the hotel will sell one less room. Using this industry standard, do the following: A) calculate the current sales revenue (income from room sales) using the price you obtained and the number of rooms. B) determine, using the optimization process, the number of $10 room rate increases that will result in the maximum sales revenue. C) what is the new room rate? D) how many rooms will be sold. And E) what is their new maximum sales revenue (income from room sales).
 
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skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
How many rooms are occupied at the given rate of \$139.99 ? All of them? Some other number < 66 ?
 
Apr 2016
6
0
Rhode Island
Yes all 66 are occupied occupied originally
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
revenue for the base rate and for added multiples of \$10 ...

R = (139.99)(66)

R = (139.99 + 10)(66 - 1)

R = (139.99 + 20)(66 - 2)

R = (139.99 + 30)(66 - 3)

see a pattern?
 
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Apr 2016
6
0
Rhode Island
Is that the only way to do it by doing each one out like that?
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
Is that the only way to do it by doing each one out like that?
I made a list so you may see a pattern and come up with a revenue function on your own ...
 
Apr 2016
6
0
Rhode Island
I see the pattern but I still don't understand what I'm supposed to do
 
Apr 2016
6
0
Rhode Island
Would it be like R(x)=(10x + 139.99)(66-x) ? And then what would I do from there?
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
You need to derive a revenue function and determine a maximum for that function using the optimization methods learned in calculus.

I've given you a place to start ... that's about all I can tell you because I'm sure it is a graded project.
 
Dec 2013
2,002
757
Colombia
Would it be like R(x)=(10x + 139.99)(66-x) ?
Yes. That's good. Now, how can you determine the maximum revenue that this function gives?