The Hotel Bellville has 400 rooms. Currently the hotel is filled . The daily rental is $ 250 per room.
For every $ 14 increase in rent the demand for rooms decreases by 5 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 ?
I don't really understand how to start this problem...Am I supposed to use the point-slope formula? And if so, how do I know which is the independent or dependent variable?
revenue = rate * #rooms
= (250+14x)(400-5x)
this will be a parabola, where max revenue is at the vertex.
thanks!
To solve this problem, we need to find the value of x that maximizes the revenue of the hotel. Let's break down the steps:
Step 1: Determine the demand equation
The demand for rooms is decreasing with each $14 increase in rent. For every $14 increase, the demand decreases by 5 rooms. So, we can use the following equation to represent the demand for rooms:
Demand = 400 - 5x, where x represents the number of $14 increases in rent.
Step 2: Determine the revenue equation
Revenue is calculated by multiplying the number of rooms rented by the rental price. In this case, the rental price is given as $250 per room. So, the revenue equation is:
Revenue = (400 - 5x) * 250
Step 3: Find the value of x that maximizes revenue
To find the value of x that maximizes revenue, we need to find the vertex of the revenue equation. The vertex of a quadratic equation in the form of ax^2 + bx + c can be found using the formula:
x = -b/2a
In our revenue equation, the coefficient of x^2 is 0, so we can disregard the quadratic term. The equation can be simplified to:
Revenue = -1250x + 100000
Comparing this equation to the general form, we can identify a = -1250 and b = 0. Plugging these values into the formula, we get:
x = -(0) / (2 * (-1250))
x = 0
Since the coefficient of x^2 is 0, the parabola is linear. It means that the vertex lies at the endpoint, which is x = 0.
Therefore, in order to maximize the revenue of the hotel, x should be 0.