a real estate office manages 50 apartments in 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 be real estate office charge to maximize profits?
let the number of $25 increases be n
new rent = 900 + 25n
number rented = 50-n
profit = (50-n)(900+25n) - 75n
= 45000+275n - 25n^2
d(profit)/dn = 275 - 50n = 0 for max/min of profit
50n = 275
n = 275/50 = 5.5
The question is not clear whether the increases have to be in exact jumps of $25.
if so, then n=5 or n=6 will yield the same profit
e.g.
if n=6, profit = 45000+1650-900 = 45750
if n=5 , profit = 45000+1375-625 = 45750
if a partial increase is allowed, then
n = 5.5
new rent = 1037.50
number rented = 50-5.5 = 44.5 ???? (makes no real sense)
but mathematically profit
= 45000+5.5(275)-25(5.5)^2 = 45756.25
I would go with either n=5 or n=6
To determine the rent amount that maximizes profits, we need to find the point where the revenue from renting out apartments is the highest.
Let's break down the problem and calculate the revenue and expenses for each scenario.
1. When the rent is $900 per month, all 50 units are occupied.
- Revenue: $900 x 50 = $45,000 per month
- Expenses: $75 x 50 = $3,750 per month
- Profit: Revenue - Expenses = $45,000 - $3,750 = $41,250 per month
2. For every $25 increase in rent, one unit becomes vacant.
- Therefore, for each $25 increase, the number of occupied units decreases by 1.
Now, let's calculate the revenue and expenses for different rent increases:
- $925 rent per month:
- Occupied Units: 49
- Revenue: $925 x 49 = $45,425 per month
- Expenses: $75 x 49 = $3,675 per month
- Profit: Revenue - Expenses = $45,425 - $3,675 = $41,750 per month
- $950 rent per month:
- Occupied Units: 48
- Revenue: $950 x 48 = $45,600 per month
- Expenses: $75 x 48 = $3,600 per month
- Profit: Revenue - Expenses = $45,600 - $3,600 = $42,000 per month
- $975 rent per month:
- Occupied Units: 47
- Revenue: $975 x 47 = $45,825 per month
- Expenses: $75 x 47 = $3,525 per month
- Profit: Revenue - Expenses = $45,825 - $3,525 = $42,300 per month
By continuing this pattern, we can observe that as the rent increases, the revenue initially increases, but as the number of occupied units decreases, the additional increase in revenue becomes smaller.
To determine the rent amount that maximizes profit, we need to find the point where the decrease in revenue from fewer occupied units is balanced by the increase in revenue from higher rent.
In this example, we see that the profit is maximized at a rent of $975 per month. So, the real estate office should charge $975 per month to maximize profits.