A 440-unit hotel is trying to determine their daily rates. From experience, the hotel manager knows that all rooms are typically occupied when the daily room rate is $92. Market research indicates that for every increase of x
dollars in the daily room rate, there will be x
rooms vacant. Each occupied room costs $22 per day to service and maintain. What should the hotel charge per day in order to maximize profit?
p = (440 - x) (92 - 22 + x) = (440 * 70) + 370 x - x^2
the max is on the axis of symmetry
... x = -b / 2a = -370 / -2 = 185
rate for max profit ... 92 + 185
Well, since I'm a Clown Bot, I must say, the hotel manager should charge $92,000 per day! Who wouldn't want to stay in a room for $92,000? Think of the luxury! Plus, with that kind of rate, the hotel can afford to have clowns running around, entertaining the guests all day long. Profit and laughter all in one package!
To determine the optimal daily room rate for maximizing profit, we need to consider the revenue from occupied rooms and the cost of servicing and maintaining those occupied rooms.
Let's represent the daily room rate as R (in dollars) and the number of vacant rooms as V. From the information given, we know that for every $1 increase in the daily room rate, there will be 1 vacant room (x dollars = x rooms vacant).
Based on this, we can write the equation for the number of vacant rooms as V = R - 92.
We also know that the total number of rooms in the hotel is 440. Therefore, the number of occupied rooms is O = 440 - V = 440 - (R - 92).
The revenue from occupied rooms can be calculated as the product of the number of occupied rooms and the daily room rate, which is R * O.
The cost of servicing and maintaining each occupied room is $22. So, the total cost for servicing and maintaining all occupied rooms can be calculated as 22 * O.
Finally, the profit can be calculated as the revenue from occupied rooms minus the cost of servicing and maintaining those rooms. Thus, the profit function is given by P = (R * O) - (22 * O).
To find the optimal daily room rate that maximizes profit, we need to determine the value of R that will maximize this profit function.
Taking the derivative of the profit function with respect to R and setting it equal to zero, we can find the critical point(s) where the maximum occurs:
dP/dR = (d/dR) [(R * O) - (22 * O)] = O - 22 * (dO/dR) = 0
Substituting the expression for O, we get:
V - 22 * (dO/dR) = 0
Substituting the expression for V, we get:
R - 92 - 22 * (dO/dR) = 0
Using the fact that (dO/dR) = -1, since for each increase of $1 in R, there will be 1 vacant room, we can solve for R:
R - 92 - 22 * (-1) = 0
R - 92 + 22 = 0
R - 70 = 0
R = 70
Therefore, the hotel should charge $70 per day in order to maximize profit.
To find the optimal daily rate that maximizes profit, we need to determine the number of occupied rooms and the associated profit for each potential daily rate. Then, we can compare the profits at different daily rates to identify the one that yields the highest profit.
Let's start by considering the relationship between the daily room rate and the number of occupied rooms. Based on the information provided, we know that when the daily rate is $92, all rooms are occupied. We also know that for every increase of x dollars in the daily rate, there will be x rooms vacant. This means that when the daily rate is $92 + x, there will be 440 - x occupied rooms.
Next, let's calculate the profit for each potential daily rate. The profit for a given daily rate can be found by subtracting the cost to service and maintain each occupied room from the total revenue generated by occupied rooms.
The total revenue generated by occupied rooms can be calculated by multiplying the number of occupied rooms by the daily room rate. Since the number of occupied rooms depends on the daily rate, we can express it as a function of the daily rate (R):
Occupied rooms = 440 - (daily rate - $92)
The revenue generated by occupied rooms, denoted as R(revenue), can then be defined as:
R(revenue) = (440 - (daily rate - $92)) * daily rate
The cost to service and maintain each occupied room is given as $22 per day. Therefore, the total cost to service and maintain occupied rooms, denoted as C(cost), can be expressed as:
C(cost) = (440 - (daily rate - $92)) * $22
Finally, we can calculate the profit (P) for each daily rate by subtracting the cost from the revenue:
P = R(revenue) - C(cost)
Now, we can calculate the profit for different daily rates and determine the optimal rate that maximizes profit:
1. Calculate the profit for a specific daily rate (R):
P(daily rate) = ((440 - (daily rate - $92)) * daily rate) - ((440 - (daily rate - $92)) * $22)
2. Repeat step 1 for different daily rates, incrementing the rate by a certain value each time.
3. Compare the profits obtained for different rates and identify the one that yields the highest profit.
By following these steps, the hotel can determine the optimal daily rate that maximizes their profit.