The Hotel Florence has 550 rooms. Currently the hotel is filled . The daily rental is $ 700 per room.
For every $ 14 increase in rent the demand for rooms decreases by 7 rooms.
Let x = the number of $ 14 increases that can be made.
What should x be so as to maximize the revenue of the hotel ?
What is the rent per room when the revenue is maximized? $
What is the maximum revenue? $
revenue is room-charge * #rooms, so
R(x) = (700+14x)(550-7x)
That's just a parabola, and you can easily find its vertex.
thank you
To solve this problem and determine the value of x that maximizes the hotel's revenue, we need to consider the relationship between the rental price, the number of rooms rented, and the revenue generated.
First, we will determine the total revenue at each level of x by using the given information. We know that the daily rental is $700 per room, and for every $14 increase in rent, the demand for rooms decreases by 7. So, we can calculate the number of rooms rented at each level of x as follows:
Number of rooms rented = 550 - (7 * x)
Next, we will calculate the total revenue generated by multiplying the rental price per room by the number of rooms rented:
Revenue = Rental price * Number of rooms rented
Revenue = $700 * (550 - 7x)
Revenue = $385,000 - $4,900x
To find the value of x that maximizes the revenue, we need to find the vertex of the revenue function. The vertex of a quadratic function in the form of y = ax^2 + bx + c can be found using the formula x = -b / (2a). In our case, the quadratic function is Revenue = - $4,900x + $385,000.
a = -4,900
b = 0
c = 385,000
x = -b / (2a) = -0 / (2 * -4,900) = 0
Therefore, the value of x that maximizes the revenue is 0. This means that there should be no increase in rent ($14 increases) to maximize revenue.
To find the rent per room when revenue is maximized, we substitute the value of x into the rental price formula:
Rent per room = $700 + ($14 * x)
Rent per room = $700 + ($14 * 0)
Rent per room = $700
Therefore, the rent per room when revenue is maximized is $700.
To calculate the maximum revenue, we substitute the value of x into the revenue formula:
Revenue = $385,000 - $4,900x
Revenue = $385,000 - $4,900 * 0
Revenue = $385,000
Therefore, the maximum revenue is $385,000.