We appreciate your visit to 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. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
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.
Thanks for taking the time to read 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. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada