The Hotel Regal has 600 rooms. Currently the hotel is filled . The daily rental is $ 250 per room.

For every $ 4 increase in rent the demand for rooms decreases by 4 rooms.
Let x = the number of $ 4 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? $

r(x) = (600-4x)(250+4x)

Now plug and chug.

To solve this problem, we need to find the value of x that maximizes the revenue of the hotel.

Let's break down the problem step by step:

1. Determine the initial rental price per room: The initial rental price is $250 per room.

2. Determine the decrease in demand for rooms: For every $4 increase in rent, the demand for rooms decreases by 4 rooms. This means that for every $4 increase, the hotel will lose 4 potential customers.

3. Calculate the number of customers at each rental price: To do this, we need to find the relationship between the rental price and the demand for rooms. Since the decrease in demand is linear, we can define the relationship as:
Demand = 600 - (x * 4)
Here, x represents the number of $4 increases in rent.

4. Calculate the revenue at each rental price: Revenue is calculated by multiplying the rental price by the number of occupied rooms. The formula for revenue is:
Revenue = (600 - (x * 4)) * (250 + (4 * x))
Here, (600 - (x * 4)) represents the number of occupied rooms, and (250 + (4 * x)) represents the rental price per room.

5. Find the value of x that maximizes the revenue: To do this, we can take the derivative of the revenue function with respect to x and set it equal to zero. Then we solve for x.

6. Calculate the rent per room when the revenue is maximized: Substitute the value of x that maximizes the revenue into the formula for rental price to find the corresponding rental price per room.

7. Calculate the maximum revenue: Substitute the value of x that maximizes the revenue into the formula for revenue to find the maximum revenue.

Let's do the math to find the answers to the question.