A hotel has 300 units. All rooms are occupied when the hotel charges $70 per day for a room. For every increase of x dollars in the daily room rate, there are x room vacant . Each occupied room costs $26 per day to service and maintain. What should the hotel charge per day in order to maximize daily profits
Let's assume that for every increase of $x in the daily room rate, there are x rooms vacant, which means there will be 300 - x occupied rooms.
The revenue generated from room charges each day would be (300 - x) * ($70 + x), as there are (300 - x) occupied rooms charging $70 + $x per day.
The cost of servicing and maintaining each occupied room is $26 per day, so the cost would be (300 - x) * $26.
The profit each day would be the revenue minus the cost, which is:
Profit = Revenue - Cost
Profit = (300 - x) * ($70 + x) - (300 - x) * $26
Profit = (300 - x) * ($70 + x - $26)
Profit = (300 - x) * ($44 + x)
To maximize daily profit, we need to find the value of x that maximizes the function (300 - x) * ($44 + x).
To do this, we can take the derivative of the profit function with respect to x and set it equal to zero, then solve for x.
d(Profit)/dx = ($44 + x) - (300 - x) = 0
$x - x + 44 - 300 = 0
2x - 256 = 0
2x = 256
x = 128
Therefore, to maximize daily profit, the hotel should increase the daily room rate by $128.
The new daily room rate would be $70 + $128 = $198.