You are the manager of an apartment complex with 50 units. When you set rent at $800/month, all apartments are rented. As you increase rent by $25/month, one fewer apartment is rented. Maintenance costs run $50/month for each occupied unit. What is the rent (in dollars) that maximizes the total amount of profit?
Let x be the number of $25 increases. Then the revenue is
(50-x)(800+25x)
The cost is 50(50-x)
profit to be maximized then is
p=(50-x)(800+25x)-50(50-x)
Well, if I were a clown manager, I would probably say that the rent that maximizes the total amount of profit is "free"! Imagine all the tenants lining up to live in your complex if it had free rent! But alas, I realize that's not a practical answer.
To tackle this problem seriously, we'll need to dive into the math side of things. Let's break it down step by step.
First, let's determine how many apartments are rented at each rent increase:
- At $800/month, all 50 apartments are rented.
- When we increase the rent by $25/month, one fewer apartment is rented. So, at $825/month, we have 49 apartments rented.
- By increasing the rent by another $25/month to $850, we have 48 rented apartments.
We can see that for each $25 increase in rent, one less apartment is rented. So, to find the maximum profit, we'll need to find the point where the decrease in renter count causes the loss in profit to outweigh the increase in rent.
Next, let's calculate the profit at each rent level:
- At $800/month, profit = (50 rented apartments * $800) - (50 apartments * $50/month maintenance cost).
- At $825/month, profit = (49 rented apartments * $825) - (49 apartments * $50/month maintenance cost).
- At $850/month, profit = (48 rented apartments * $850) - (48 apartments * $50/month maintenance cost).
Continuing this pattern, you can calculate the profit for each rent increase and identify the point where the profit starts decreasing. That will give you the optimal rent that maximizes total profit.
Remember, finding the optimal rent is crucial for a successful apartment complex, but remember to balance it with happy tenants too!
To find the rent that maximizes the total amount of profit, we need to analyze the relationship between the number of rented apartments and the profit. We can start by creating a table to understand the relationship between rent, the number of rented apartments, and profit.
Rent | Rented Apartments | Profit
-----------------------------------
$800 | 50 | -
$825 | 49 | -
$850 | 48 | -
$875 | 47 | -
... | ... | -
From the given information, we know that when rent is set at $800/month, all 50 apartments are rented. As the rent increases by $25/month, one fewer apartment is rented.
To calculate the profit, we need to take into account the number of rented apartments and the maintenance costs per occupied unit.
Profit = (Rented Apartments * Rent) - (Rented Apartments * Maintenance Cost)
Given that the maintenance cost is $50/month for each occupied unit, we can calculate the profit for each scenario in the table:
Rent | Rented Apartments | Profit
-----------------------------------
$800 | 50 | $22,500
$825 | 49 | $22,800
$850 | 48 | $23,100
$875 | 47 | $23,350
... | ... | ...
To find the rent that maximizes the total amount of profit, we need to identify the point with the highest profit. In this case, we can observe that as the rent increases, the profit also increases.
From the table, we can see that the profit increases by $300 for each reduction in the number of rented apartments. Therefore, we could continue calculating the profit until no apartments are rented:
Rent | Rented Apartments | Profit
-----------------------------------
$800 | 50 | $22,500
$825 | 49 | $22,800
$850 | 48 | $23,100
$875 | 47 | $23,350
$900 | 46 | $23,600
... | ... | ...
However, since we want to find the rent that maximizes profit and not necessarily the profit for each scenario, we can take a different approach.
We can calculate the profit for each scenario and observe how the profit changes. From the given information, we know that when rent is set at $800/month, all 50 apartments are rented. As the rent increases by $25/month, one fewer apartment is rented.
At $800/month, all apartments are rented, and the profit is $22,500. Now, let's calculate the profit when one fewer apartment is rented:
Profit with one fewer rented apartment:
Profit = (49 * $825) - (49 * $50) = $22,275
By comparing the two scenarios, we can see that increasing the rent by $25/month while reducing the number of rented apartments results in a decrease in profit of $225.
Continuing this analysis, we can observe that for each reduction in the number of rented apartments, the profit will decrease by $225.
To find the rent that maximizes profit, we need to determine at which point the profit starts to decrease. In this case, we can see that when rent is increased to $850/month, the profit remains the same as in the previous scenario ($22,275).
Therefore, the rent that maximizes the total amount of profit is $850/month.
To find the rent that maximizes the total profit, we need to consider the relationship between the rent, the number of rented apartments, and the total profit.
Let's start by breaking down the problem into steps:
Step 1: Identify the relevant variables:
- Rent per month: denoted as R (in dollars)
- Number of rented apartments: denoted as A
- Total profit: denoted as P (in dollars)
Step 2: Determine the relationship between the variables:
- From the given information, we know that when the rent is $800/month (R = 800), all apartments are rented (A = 50).
- Additionally, for every $25 increase in rent, one fewer apartment is rented.
Based on this information, we can establish the relationship between R and A as:
A = 50 - (R - 800)/25
Step 3: Calculate the profit (P):
- The total profit can be calculated by subtracting the total maintenance cost from the total revenue. Given that the maintenance cost is $50/month per occupied unit, the total revenue is Rent per month (R) multiplied by the Number of rented apartments (A), and the total maintenance cost is $50 multiplied by the Number of rented apartments (A).
So, P = (R * A) - (50 * A)
Step 4: Express P in terms of R:
Substituting the value of A from the relationship (A = 50 - (R - 800)/25) into the equation for P, we have:
P = (R * (50 - (R - 800)/25)) - (50 * (50 - (R - 800)/25))
Step 5: Simplify the equation for P:
Simplifying the equation for P will give us a polynomial expression, which we can then maximize to find the rent that maximizes profit.
P = (R * (50 - (R - 800)/25)) - (50 * (50 - (R - 800)/25))
Simplifying further:
P = (R * (50 - (R - 800)/25)) - (50 * (50 - (R - 800)/25))
P = (50R - 2(R - 800)) - (50(50) - 2(R - 800))
P = 50R - 2R + 1600 - 50(50) + 2(R - 800)
P = 48R - 2R + 1600 - 2500 + 2R - 1600
P = -2R - 900
Step 6: Maximize the profit:
To maximize the profit, we need to find the value of R that maximizes the polynomial expression P = -2R - 900.
Since the coefficient of R is negative, the value of R that maximizes P is when R approaches negative infinity (as R increases without bound). Thus, there is no specific rent value that maximizes the profit.
However, if you are looking for the maximum possible profit, you can plug in the given rent values to find the corresponding profits and select the largest one.