Mr. Frank Graham has recently assumed ownership of a historic hotel in Lehi, UT. The hotel is located a little outside of town, surrounded by the natural beauty of Cache Valley mountains, and is only a short drive away from Thanksgiving Point, a museum that attracts tourists. Last year, Mr. Graham’s investment firm completed renovations to convert the historic property into a luxury resort. The property now has 80 rooms.

Mr. Graham needs to set the nightly rate so that he can begin to turn a profit and repay the investors. Maintenance costs for an occupied room average $4 per day, which includes staff wages, supplies, and utilities. Some of the local business owners have told Mr. Graham that a good rule of thumb for the non-holiday season is that for every $1 increase in the nightly rate, one less room will be rented. The last time all of the rooms were occupied (other than the holidays) was when the nightly rate was $60 per room.

Mr. Graham would like to know how much he should charge per room in order to maximize his profit and what his profit would be at that rate. Additionally, he would like a procedure for finding the daily rate that would maximize his profit in the future, even if the hotel prices and maintenance costs change.

Task:

You are required to write a report of your mathematical models explaining the procedure for finding the daily rate that would maximize his profit in the future in two ways:

1. Techniques from algebra.
2. Techniques from calculus.

To receive full credit, please do the following:
1. Show all your work clearly.
2. Provide justifications for each of your models.
3. Explicitly state any simplifying assumptions made during your mathematical decisions.

Extension 1: Booking Fees

Mr. Graham has the option to advertise his hotel on travel sites like hotels.com. While advertising might bring in new guests, the booking fees are typically 3% of the room price. The booking fee can be absorbed by the guest by adding it to the nightly rate. How much should he charge per room in order to maximize his profit, and what is his profit at that rate?

Extension 2: Saving for Renovations

Additionally, Mr. Graham would like to set aside some of the revenue, around 3%, to pay for future major maintenance and renovations. How much should he charge per room in order to maximize his profit, and what is his profit at that rate?

Mr. Graham should set the room rate at $70 per night to maximize his profit. The corresponding maximum daily profit would be $P(10) = (80-10)(60+10) - 4(80-10) = $4620.

To maximize his profit, Mr. Graham should set the nightly rate at $70. This decision is based on a profit function, [tex]$ P(x) = (80-x)(60+x) - 4(80-x) $[/tex],

where [tex]$ x $[/tex] represents the increase in room rate.

This function is derived from the assumption that for every $1 increase in the room rate, one less room will be rented. Starting with all 80 rooms rented at $60 each, each $1 increase in rate results in one less room rented.

Therefore, if the rate is increased by $x, there will be [tex]$ (80-x) $[/tex] rooms rented at [tex]$ (60+x) $[/tex] dollars.

To find the optimal rate, we expand the profit function to[tex]$ P(x) = 4800 + 20x - x^2 $[/tex] and calculate the vertex of the parabola, which represents maximum profit.

This is done using the formula [tex]$ x = -b/(2a) $[/tex], where [tex]$ a = -1 $[/tex] and [tex]$ b = 20 $[/tex]. This calculation yields [tex]$ x = 10 $[/tex], indicating that the maximum profit occurs when the room rate is increased by $10 to $70 per night.

Alternatively, using calculus, we find the derivative of the profit function, [tex]$ P'(x) = 20 - 2x $[/tex], set it equal to zero to find critical points, and solve for [tex]$ x $[/tex]. which again gives [tex]$ x = 10 $[/tex]. Substituting [tex]$ x = 10 $[/tex] back into the profit function confirms that the maximum profit occurs when the room rate is $70 per night, with a maximum daily profit of $4620.

Therefore, Mr. Graham should set the room rate at $70 per night to maximize his profit. The corresponding maximum daily profit would be $P(10) = (80-10)(60+10) - 4(80-10) = $4620

Question:-

Mr. Frank Graham has recently assumed ownership of an historic hotel in Lehi, UT. The hotel is located a little outside of town surrounded by the natural beauty of Cache Valley mountains and only a short drive away from Thanksgiving Point, a museum that attracts tourists. Last year, Mr.

Graham’s investment firm completed renovations to convert the historic property into a luxury resort. The property now has 80 rooms. Mr. Graham needs to set the nightly rate so that he can begin to turn a profit and repay the investors. Maintenance costs for an occupied room average $4 per day, which includes staff wages, supplies, and utilities. Some of the local business owners have told Mr. Graham that a good rule of thumb for non-holiday season is that for every $1 increase in the nightly rate, one less room will be rented. The last time all of the rooms were occupied (other than the holidays) was when the nightly rate was $60 per room.

Mr. Graham would like to know how much he should charge per room in order to maximize his profit and what his profit would be at that rate. Also, he would like to have a procedure for finding the daily rate that would maximize his profit in the future, even if the hotel prices and maintenance costs change. You are required to write a report of your mathematical models explaining the procedure finding the daily rate that would maximize his profit in the future in two ways: (1) Techniques from algebra and (2) Techniques from calculus. To receive full credit please do the following:

1. Please show all your work clearly.

2. Provide justifications for each of your models.

3. Please explicitly state any simplifying assumptions you made during your mathematical decisions.

Extension 1. Booking fees.

Mr. Graham has the option to advertise his hotel on travel sites like hotels.com. While advertising might bring in new guests, the booking fees are typically 3% of the room price. The booking fee can be absorbed by the guest by adding it on to the nightly rate. How much should he charge per room in order to maximize his profit and what is his profit at that rate?

Extension 2. Saving for renovations.

Additionally, Mr. Graham would like to set aside some of the revenue, around 3%, to pay for future major maintenance and renovations. How much should he charge per room in order to maximize his profit and what is his profit at that rate?