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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 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:

1. How much should he charge per room in order to maximize his profit?
2. What would his profit be at that rate?
3. 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:**

Write a report on 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 do the following:

1. Show all your work clearly.
2. Provide justifications for each of your models.
3. Explicitly state any simplifying assumptions you made during your mathematical decisions.

Answer :

To maximize profit, Mr. Graham should charge $70 per night. This is found by creating a profit function $P(x) = (80-x)(60+x) - 4(80-x)$, where $x$ is the increase in room rate, and maximizing it using either algebraic methods (finding the vertex of the parabola) or calculus methods (setting the derivative to zero). The assumptions made are a linear relationship between price increase and occupancy decrease, fixed maintenance costs, and no other revenues or costs.

Profit Maximization for Mr. Graham's Hotel

Mr. Graham needs to determine the optimal nightly rate for his hotel to maximize profit, considering that for every $1 increase in room rate, one less room will be rented. Assuming all 80 rooms are rented at $60 each, as the baseline, each $1 increase will result in losing one occupancy. So, if the room rate is increased by $x, there will be $(80-x)$ rooms rented at $(60+x)$ dollars.

The total revenue is $R(x) = (80-x)(60+x)$, and the cost for the occupied rooms is $C(x) = 4(80-x)$, since unoccupied rooms do not incur maintenance costs. The profit function would then be $P(x) = R(x) - C(x) = (80-x)(60+x) - 4(80-x)$.

To maximize profit using algebraic techniques, we can expand the profit function, $P(x) = 4800 + 20x - x^2$, and calculate the vertex of the parabola (which represents maximum profit), where $x = -b/(2a)$ from the general form $ax^2 + bx + c$.

In this case, $a=-1$ and $b=20$, so the optimal increase in rate is $x = -20/(2(-1)) = 10$. Hence, the maximum profit will be when the room rate is $60 + 10 = $70 per night.

For calculus methods, we take the derivative of the profit function, $P'(x) = d/dx(4800 + 20x - x^2) = 20 - 2x$, and set it equal to zero to find the critical points. Solving $P'(x) = 0$ gives $x = 10$, confirming the algebraic result. Substituting $x=10$ into the profit function gives us the maximum profit.

Assumptions:

  • The relationship between room rate increase and occupancy decrease is linear and consistent.
  • All rooms are equivalent and maintenance costs are fixed.
  • There are no other costs or sources of revenue.

Conclusion

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

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