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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 the 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 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 rooms were occupied (other than holidays) was when the nightly rate was $60 per room.

**Task:**

Mr. Graham would like to know:

1. How much he should charge per room to maximize his profit and what his profit would be at that rate.
2. A procedure for finding the daily rate that would maximize his profit in the future, even if hotel prices and maintenance costs change.

**Requirements:**

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 using:

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

To receive full credit, please:

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.

**Extensions:**

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 to maximize his profit, and what is his profit at that rate?

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 to maximize his profit, and what is his profit at that rate?

Answer :

To maximize profit, set nightly rate to $62. Profit at this rate is approximately $3368. Calculated using quadratic equation optimization.

Mathematical Model using Algebra:

Let's denote the nightly rate as [tex]\( R \)[/tex] and the number of rooms rented as [tex]\( n \)[/tex].

Given:

- Maintenance cost per occupied room = $4

- Last time all rooms were occupied, the nightly rate was $60

- For every $1 increase in the nightly rate, one less room will be rented

Based on the given information, we can write the following equations:

1. Total revenue [tex](\( \text{TR} \)) = \( R \times n \)[/tex]

2. Total maintenance cost [tex](\( \text{MC} \))[/tex] = $4 per occupied room [tex]\( \times \)[/tex]number of occupied rooms [tex](\( n \))[/tex]

Profit [tex](\( \text{P} \))[/tex] = Total revenue [tex](\( \text{TR} \))[/tex] - Total maintenance cost [tex](\( \text{MC} \))[/tex]

We know that for every $1 increase in the nightly rate, one less room will be rented. Therefore, we can write the relationship between [tex]\( R \)[/tex] and [tex]\( n \)[/tex] as:

[tex]\[ R = 60 + (60 - n) \][/tex]

Substituting this relationship into the profit equation:

[tex]\[ \text{P} = (60 + (60 - n)) \times n - 4n \][/tex]

[tex]\[ \text{P} = (60n + 60n - n^2) - 4n \][/tex]

[tex]\[ \text{P} = 120n - n^2 - 4n \][/tex]

[tex]\[ \text{P} = -n^2 + 116n \][/tex]

To maximize profit, we find the value of [tex]\( n \)[/tex] that maximizes the quadratic equation [tex]\( -n^2 + 116n \)[/tex]. We can find the vertex of this quadratic equation using the formula:

[tex]\[ n = \frac{-b}{2a} \][/tex]

where [tex]\( a = -1 \)[/tex] and [tex]\( b = 116 \)[/tex].

[tex]\[ n = \frac{-116}{2 \times (-1)} \][/tex]

[tex]\[ n = \frac{-116}{-2} \][/tex]

[tex]\[ n = 58 \][/tex]

Now, we can find the corresponding value of [tex]\( R \)[/tex] using the relationship [tex]\( R = 60 + (60 - n) \)[/tex]:

[tex]\[ R = 60 + (60 - 58) \][/tex]

[tex]\[ R = 62 \][/tex]

Therefore, Mr. Graham should charge $62 per room to maximize his profit, and his profit at that rate is [tex]\( -58^2 + 116 \times 58 = \$3368 \)[/tex].

Justifications:

- We used algebraic techniques to model the relationship between the nightly rate and the number of rooms rented.

- We formulated the profit equation and then maximized it to find the optimal nightly rate for maximizing profit.

- Assumption: We assumed that the relationship between the nightly rate and the number of rooms rented is linear.

Extensions 1 and 2 can be similarly solved by adjusting the profit equation to account for booking fees and savings for renovations, respectively.

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