A real estate company owns 218 efficiency apartments, which are fully occupied when the rent is $940 /month. The company estimated that for each $25 increase in rent, 5 apartments will become unoccupied. What rent should be charged so that the company will receive the maximum monthly income?

To determine the rent that should be charged to maximize the monthly income, we can create a table to analyze the relationship between the rent and the number of occupied apartments:

Rent: $940, Occupied Apartments: 218
Rent: $965, Occupied Apartments: 213
Rent: $990, Occupied Apartments: 208
Rent: $1015, Occupied Apartments: 203
Rent: $1040, Occupied Apartments: 198
Rent: $1065, Occupied Apartments: 193
Rent: $1090, Occupied Apartments: 188
Rent: $1115, Occupied Apartments: 183
Rent: $1140, Occupied Apartments: 178
Rent: $1165, Occupied Apartments: 173
Rent: $1190, Occupied Apartments: 168
Rent: $1215, Occupied Apartments: 163
Rent: $1240, Occupied Apartments: 158
Rent: $1265, Occupied Apartments: 153
Rent: $1290, Occupied Apartments: 148
Rent: $1315, Occupied Apartments: 143
Rent: $1340, Occupied Apartments: 138
Rent: $1365, Occupied Apartments: 133
Rent: $1390, Occupied Apartments: 128
Rent: $1415, Occupied Apartments: 123
Rent: $1440, Occupied Apartments: 118
Rent: $1465, Occupied Apartments: 113
Rent: $1490, Occupied Apartments: 108
Rent: $1515, Occupied Apartments: 103
Rent: $1540, Occupied Apartments: 98
Rent: $1565, Occupied Apartments: 93
Rent: $1590, Occupied Apartments: 88
Rent: $1615, Occupied Apartments: 83
Rent: $1640, Occupied Apartments: 78
Rent: $1665, Occupied Apartments: 73
Rent: $1690, Occupied Apartments: 68
Rent: $1715, Occupied Apartments: 63

As we can see, the number of occupied apartments decreases by 5 for every $25 increase in rent. Therefore, we can determine the rent that results in zero occupied apartments by dividing the difference in rent ($1715 - $940 = $775) by the decrease in occupied apartments per $25 ($5).
$$\frac{775}{5} = 155$$

Using the formula
\[\text{Rent} = \text{Base rent} + (\text{Number of $25 increases} \times 25)\]

we can determine the rent that will result in zero occupied apartments:

\[\text{Rent} = \$940 + (155 \times 25)\]
\[\text{Rent} = \$940 + \$3875\]
\[\text{Rent} = \$4815\]

Thus, to maximize the monthly income, the company should charge a rent of $4815.

To find the rent that will result in the maximum monthly income for the real estate company, we need to determine the point at which the number of occupied apartments multiplied by the rent amount is the highest.

Let's start by understanding the relationship between rent and the number of occupied apartments.

From the given information, we know that when the rent is $940/month, all 218 apartments are fully occupied. For each $25 increase in rent, 5 apartments become unoccupied.

So, we can establish the following relationship between the change in rent ($25) and the change in the number of occupied apartments (+5):

Change in Rent / Change in Occupied Apartments = $25 / 5 = $5 per occupied apartment.

Now, we can determine the rent that maximizes the company's monthly income. We'll use a trial and error method:

1. Start with the initial rent of $940/month.
Number of occupied apartments = 218
Total monthly income = Rent * Number of occupied apartments = $940 * 218

2. Increase the rent by $25 and calculate the new monthly income.
Number of occupied apartments = 218 - (5 * 1) = 213
Total monthly income = (Rent + $25) * Number of occupied apartments

3. Repeat this process, increasing the rent by $25 each time and calculating the monthly income until the number of occupied apartments becomes zero.

4. Select the rent amount that results in the maximum monthly income.

Let's calculate the monthly income for a few trial rents:

Rent: $940/month
Number of occupied apartments = 218
Total monthly income = $940 * 218 = $204,920

Rent: $965/month
Number of occupied apartments = 213
Total monthly income = $965 * 213 = $205,545

Rent: $990/month
Number of occupied apartments = 208
Total monthly income = $990 * 208 = $205,920

Rent: $1,015/month
Number of occupied apartments = 203
Total monthly income = $1,015 * 203 = $206,045

By comparing the monthly incomes, we can observe that as the rent increases, the total monthly income initially increases but then starts to decrease after a certain point. The maximum monthly income occurs when the number of occupied apartments becomes zero.

Therefore, the rent that should be charged to maximize the company's monthly income is $990/month. At this rent amount, the company will earn a total monthly income of $205,920.

let the number of $25 increases be n

rent = 940 + 25n
number rented = 218-5n

income
= (940+25n)(218-5n)
= 204920 +750n - 125n^2

If you know Calculus:
d(income) = 750 - 250n = 0 for a max of income
250n=750
n = 3

if you don't know Calc
complete the square:
-125(n^2 - 6n + 9 - 9) + 204920
= -125( (n-3)^2 - 9) + 204920
= -125(n-3)^2 + 1125 + 204920
= -125(n-3)^2 + 206045
vertex is (3, 206045)

there should be 3 increases for a rent of 940+75 = 1015

or
the x of the vertex is -b/(2a) = -750/-250 = 3
for 3 increases.