A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. For every $25 increase in rent, one unit becomes vacant. On average, all units require $75 in maintenance and repairs each month. How much rent should the real estate office charge to maximize profits?
I don't even know the answer to this one. I don't understand at all, please help me step by step
right now:
price of rent = 900
number of units rented = 50
Let the number of $25 increases be n
(e.g. If n= 2 , new rent is 900+2(25) = 950
if n = 5 , the new rent 900 + 5(25) = 1025
so the new rent = 900 + 25n
number rented = 50-n
maintenance cost = 75(50-n)
Profit = P = (900+25n)(50-n) - 75(50-n)
= 45000 + 350n - 25n^2 - 3750 + 75n
= 41250 + 425n - 25n^2
d(profit)/dn = 425 - 50n
= 0 for a max of P
50n = 425
n = 8.5
The question did not say if increases are in whole multiples of 25 , but I will assume that. We could not rent 50-8.5 or 41.5 units.
when n = 8 or n = 9
if n = 8
number rented = 42
rent = 900+8(25) = 1100
maintenace cost = 75(42) = 3150
Profit = 42x1100 - 3150 = 43050
if n = 9
number rented = 41
rent = 1125
maintenance cost = 3075
Profit = 41x1125 - 3075 = 43050 , as expected.
Wow thankyou sooooo much! you saved me xD
I'm interested in multi-unit buildings from 8 214 maybe just 20 apartment buildings if you do call me please leave me a voicemail because I'm a professional boxer and people call me from all around the world and you could send me a link awesome to my email. Luis_mateo12. my cell number 773 680 5343 please leave me a voicemail and let me know as you and I will definitely call you back because a lot of people just call me for nonsense
To find the rent that maximizes profits for the real estate office, we need to analyze the relationship between rent, occupancy, and expenses.
Step 1: Identify the key variables:
- Rent: The amount charged for each apartment unit per month.
- Occupancy: The number of apartments that are occupied.
- Maintenance and repairs: The monthly expenses required to maintain and repair all the units.
- Profit: The income generated by the real estate office after deducting expenses.
Step 2: Understand the given information:
- When the rent is $900 per month, all 50 units are occupied.
- For every $25 increase in rent, one unit becomes vacant.
- All units require $75 in maintenance and repairs each month.
Step 3: Determine the relationship between rent and occupancy:
- From the given information, we can deduce that for every $25 increase in rent, the number of occupied units decreases by one. This means that the occupancy is decreasing linearly with the increase in rent.
- The initial rent ($900) corresponds to full occupancy (50 units occupied).
- For every $25 increase in rent, one unit becomes vacant, resulting in 50 - (25 * k) occupied units, where k represents the number of $25 rent increments.
Step 4: Calculate the occupancy at any given rent:
- If the initial rent ($900 per month) corresponds to 50 units occupied, we can determine the occupancy at any given rent using the relationship identified above.
- Occupancy = 50 - (25 * k)
Step 5: Calculate the total income generated at any given rent:
- Total income = Rent * Occupancy
- Since every occupied unit generates the same rental income, we multiply the rent by the occupancy.
Step 6: Calculate the total expenses at any given rent:
- Total expenses = (Maintenance and repairs) * (Number of occupied units)
- As all units require $75 in maintenance and repairs each month, we multiply this by the number of occupied units to get the total expenses.
Step 7: Calculate the profit at any given rent:
- Profit = Total income - Total expenses
Step 8: Create a table to analyze the relationship between rent, expenses, and profit:
| Rent | Occupancy | Total Income | Total Expenses | Profit |
|------|-----------|--------------|----------------|--------|
| 900 | 50 | 45,000 | 3,750 | 41,250 |
| 925 | 49 | 45,425 | 3,675 | 41,750 |
| 950 | 48 | 45,600 | 3,600 | 42,000 |
| ... | ... | ... | ... | ... |
By using the formulas and table above, you can calculate the total income, expenses, and profit at different rent levels. The maximum profit occurs when the rent is determined to be the highest value that still results in a positive profit. Find the rent level where the profit is maximized and choose that as the optimal rent for the real estate office to charge.