The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 322 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 7 dollar increase in rent. Similarly, one additional unit will be occupied for each 7 dollar decrease in rent. What rent should the manager charge to maximize revenue

To determine the rent that would maximize revenue, we need to find the price point that maximizes the number of occupied units.

Let's break down the given information:

- The manager knows that if the rent is $322 per month, 110 units will be occupied.
- According to the market survey, for each $7 increase in rent, one additional unit will remain vacant.
- Similarly, for each $7 decrease in rent, one additional unit will be occupied.

To find the optimal rent, we'll follow these steps:

1. Assign variables:
- Let's call the rent R (in dollars).
- Let N represent the number of units occupied.

2. Determine the relationship between rent and the number of occupied units:
- From the given information, we can infer that the number of occupied units (N) varies inversely with an increase or decrease in rent.

3. Write an equation representing the relationship:
- Based on the information, we can write the equation as follows:
N = 110 + (7 * (322 - R) / 7)

Breaking it down:
- 322 - R represents the change in rent from $322.
- (7 * (322 - R) / 7) represents the number of additional units to be occupied or vacant, based on the change in rent.

4. Simplify the equation:
- By simplifying the equation, we get: N = 110 + (322 - R)

5. Calculate the total revenue:
- To find the revenue, we need to multiply the rent (R) by the number of occupied units (N).
- Revenue = R * N.
- Substituting the simplified equation: Revenue = R * (110 + (322 - R))

6. Expand and simplify the equation:
- Revenue = (110R + (322R - R^2))

7. Determine the rent that maximizes revenue:
- To find the rent that maximizes revenue, we need to find the value of R that yields the maximum revenue.
- This can be done by finding the vertex of the quadratic equation.
- For the equation Revenue = (110R + (322R - R^2)), the vertex can be found using different methods, such as completing the square or using calculus.

8. Solve for the vertex:
- Depending on the method used, we find that the vertex occurs at R = 216.
- Hence, the rent that maximizes revenue is $216 per month.

Therefore, the manager should charge $216 per month to maximize the revenue.