A company manages an apartment building with 126 units. Experience has shown that if the rent for each of the units is $1200 a month, a total of 84 units will be rented, and for each $50 increase in the monthly rent, 3 fewer units will be rented. Find the maximum revenue.
If there are x $50 increases, then the revenue will be
(84-3x)(1200+50x)
Now just find the vertex of that parabola.
To find the maximum revenue, we need to determine the rent amount that will yield the maximum number of rented units. Let's break down the problem step by step:
1. Let's assume the monthly rent for each unit is x dollars.
2. We know that at $1200 rent, 84 units will be rented. From this, we can calculate the number of rented units for any given rent amount.
3. The decrease in the number of rented units for every $50 increase in the rent is given as 3 units. Using this information, we can determine the number of rented units for any given rent amount.
4. Finally, we can calculate the maximum revenue by multiplying the number of rented units with the corresponding rent amount.
Let's find the optimal rent amount and maximum revenue:
1. Start with the given information: Rent at $1200/month leads to 84 rented units.
2. For every $50 increase in rent, there will be 3 fewer rented units. So, the decrease in rented units per $1 increase in rent will be 3/50.
3. To calculate the number of rented units for any given rent (x), we can use the following equation:
rented_units = 84 - (x - 1200) * (3/50)
This equation is derived by multiplying the decrease in rented units per $1 increase by the difference between the given rent (x) and $1200, and subtracting this from the initial 84 rented units.
Now, we can find the rent value that maximizes the number of rented units:
4. By setting the rented_units equation equal to zero, we can solve for the rent amount at which no units will be rented:
84 - (x - 1200) * (3/50) = 0
Simplifying this equation:
84 - (x - 1200) * (3/50) = 0
84 = (x - 1200) * (3/50)
168 = 3x - 3600
3x = 3768
x = 1256
This means that if the rent is set at $1256 per month, no units will be rented. Therefore, to maximize the rented units, the rent needs to be set below $1256.
5. To find the maximum revenue, we multiply the rent amount (x) by the number of rented units:
revenue = x * rented_units
revenue = x * (84 - (x - 1200) * (3/50))
Substitute the value of x we found earlier:
revenue = $1256 * (84 - ($1256 - 1200) * (3/50))
Solve this equation to find the maximum revenue.
By following these steps, you can calculate the maximum revenue by finding the optimal rent amount that maximizes the number of rented units.