Applications of the Derivative

[RoxieApoxie]

I'm having a bit of trouble with this whole thing, because I'm really not sure what they are asking. I think with letter A, you have to multiply the price per day by the occupancy rate to find the answer, times 200? But I'm really not sure. I can't even find a starting point. I can find derivatives, and plug in numbers just fine - could somebody give me some clues on all of these so I can figure out what I'm supposed to be doing? Thank you so much for your help!

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The data in the table below, from a survey of resort hotels with comparable rates on Hilton Head Island, show that room occupancy during the off-season (Nov-Feb) is related to the price charged for a basic room.


Price Per Day Occupancy Rate,%
$69 53
$89 47
$95 46
$99 45
$109 40
$129 32
The goal is to use these data to help answer the following questions.

A. What Price per day will maximize the daily off-season revenue for a typical hotel in the is group if it has 200 rooms available?

B.Suppose that for this typical hotel the daily cost is $4592plus $30 per occupied room. What price will maximize the profit for this hotel in the off season?

The price per day that will maximize the off-season profit for this typical hotel applies to this group of hotels. To find the room price per day that will maximize the daily revenue and the room price per day that will maximize the profit for this hotel (and thus the group of hotels) in the off-season, complete the following.

1. Multiply each occupancy rate by 200 to get the hypothetical room occupancy. Create the revenue data points that compare the price with revenue, R, which is equal to price times the room occupancy

2. Use your calculator to create an equation that models the revenue, R, as a function of the price per day, x.

3. Use maximization techniques to find the price that these hotels should charge to maximize daily revenue.

4. Use your calculator to get the occupancy as a function of the price, and use the occupancy function to create a daily cost function.

5. Form the profit function

6. Us maximization techniques to find the price that will maximize the profit.

 

[JeffM]

When you are the LEAST bit stuck on applying math to a problem, start by assigning symbols to relevant concepts (what tkhunny calls "naming things") and by writing down the known relationships for those symbols.

In this problem you are asked to find relationships between charge per room per day = C, occupancy rate = R, number of rooms occupied = N, revenue or gross income = G, expense = E, and profit = P.

You are asked 2 questions and given a whole list of hints. The first one gives the relationship between O and N, namely
N= 200R.
What is the relationship between G and N?
What is the relationship among G, E, and P?
You are started. What comes next?

 

[RoxieApoxie]

Ok, so I think I've figured parts A and B out.

We're looking for the price per day that will maximize the revenue if there is 200 rooms available.

So with your help, you helped me with the equation N=200R so I plugged in and solved, then next I had to find some numbers.

So here's what I came to:

106x69=7314
94x89=8366
92x95=8740
90x99=8910
80x109=8720
64x129=8256

therefore the price per day that will max rev is $99 per day.

and letter B states that the hotel daily cost is $4592 plus $30 per occupied room

So I set up another equation where x=rooms occupied, and y=prices per day

So, my actual equation is x(30)+x(y)+4592 = max profit

So I plugged in all those numbers and came to the max profit to also be $99.

Am I on the right track so far, or did I just do it wrong anyway?

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So for #1, am I creating a scatter plot of this data after I solve 200 x Occupancy Rate for each one? Then plotting those numbers with rooms actually filled by the price?

But for #2, it says use a calculator to create an equation - how do I do that from a graph, I believe I have to use the quadratic function formula which is y=ax^2+bx+c, but how do I get there from a graph?

Number 3, I'm assuming you just use whatever formula you come up with and find it's derivative - but what do I plug in to get the max rev?

Really unclear about number 4 thru 6 though.

Any more help would be awesome. Thanks for getting me started here

 

[RoxieApoxie]

Ok, for number two - I used the plots of (106,7314), (94,8360), (92,8740), (90,8910), (80,8720), (64,8256) and plugged them into my calculator using scatter plots and using the Quad Reg feature on my TI 83 Plus.

So for the equation I came up with R(x) = -2.49x^2+403.42x-7393.03

I feel that having such big numbers makes this very wrong, is it?

 

[JeffM]

I set up the following notation

charge per room per day = C,
occupancy rate = R,
number of rooms occupied = N,
revenue or gross income = G,
expense = E, and
profit = P.

You said

Ok, so I think I've figured parts A and B out.

We're looking for the price per day that will maximize the revenue if there is 200 rooms available.

So with your help, you helped me with the equation N=200R so I plugged in and solved, then next I had to find some numbers.

So here's what I came to:

106x69=7314
94x89=8366
92x95=8740
90x99=8910
80x109=8720
64x129=8256 These look like good arithmetic, but I am doing this while riding in the car so don't rely on MY arithmetic. In any case, arithmetic is premature.

therefore the price per day that will max rev is $99 per day. This is not necessarily so. At the prices that you have been given, this is the revenue maximizing price, but there MAY be another price that you were not given that produces an even higher revenue. To determine that you must use the given data to ESTIMATE a revenue function. We can't estimate maximum revenue without a revenue function. Nor can we estimate maximum profit without a profit function. But revenue is an element of profit so we are going to need a revenue function as part of the profit function. Make sense?

Now what two variables determine revenue, G in my notation?
Do you have any data that you can use to build a relationship between those two variables?
Sure you do.
I am away from home and cannot check your regressions. But you need to do a scatter plot of C and N, and do a sensible regression that gives C as a function of N. Do you see why you need N as the independent variable rather than C?