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?