An apartment building has 30units.if all the units are inhabited,the rent for each is 700$per unit.for every empty unit,management increases the remaining rent by 25$.find the maximum profit
if there are x empty units, the revenue is
(30-x)(700+25x)
The vertex of this parabola is at x = 1, making the rent $21,025
To find the maximum profit, we need to determine the number of occupied units and the number of empty units. Let's assume x represents the number of empty units.
If all 30 units are occupied, the rent for each unit is $700. So the total rent from occupied units will be 30 units x $700/unit = $21,000.
For every empty unit, the management increases the remaining rent by $25. So if there are x empty units, the rent increase will be x units x $25/unit = $25x.
This means the total rent from occupied units will be decreased by $25x.
The total profit will be the total rent from occupied units minus the rent decrease:
Profit = Total rent from occupied units - Rent decrease
= $21,000 - $25x.
To find the maximum profit, we need to determine the value of x which maximizes this equation. Since we are maximizing profit, we want to maximize the term $25x. Therefore, we need to minimize the value of x.
Since we have 30 units in total, the maximum number of empty units is 30. In this case, x = 30.
Substituting this value into the equation, we get:
Profit = $21,000 - $25(30)
= $21,000 - $750
= $20,250.
Therefore, the maximum profit that can be obtained is $20,250.
To find the maximum profit, we need to determine the number of empty units that will yield the highest profit. Here's how we can calculate it:
1. Let's assume the number of empty units is "x".
2. The rent for each unit, if all units are inhabited, is $700.
3. For each empty unit, the management increases the rent for the remaining units by $25.
4. Therefore, the rent for each unit, if there are "x" empty units, will be $700 + ($25 * x).
5. The total rent collection is equal to the number of inhabited units multiplied by the rent per unit.
Now, let's calculate the total revenue based on the number of empty units.
Let's assume "x" empty units.
The rent per unit, if there are x empty units, will be:
Rent per unit = $700 + ($25 * x)
The number of inhabited units will be:
Number of inhabited units = Total units - Empty units
Number of inhabited units = 30 - x
The total revenue can be calculated as:
Total Revenue = Number of inhabited units * Rent per unit
Total Revenue = (30 - x) * ($700 + ($25 * x))
Now, we can calculate the maximum profit by subtracting the total cost from the total revenue.
Total Cost = Rent per unit * Total units
Total Cost = ($700) * 30
Maximum Profit = Total Revenue - Total Cost
Maximum Profit = (30 - x) * ($700 + ($25 * x)) - ($700) * 30
To find the maximum profit, we can plot the profit equation on a graph and find the value of "x" that maximizes it. Alternatively, we can calculate the profit for each value of "x" and choose the maximum value.
I hope this helps you understand how to calculate the maximum profit for the apartment building. Feel free to ask if you have any further questions!