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To solve this problem, let’s break down the situation and use variables to form an equation for revenue based on rent.

### Step-by-Step Solution

1. **Define Variables:**
   - Let $ x $ represent the number of $8 increments (either increases or decreases) applied to the rent.
   - Define the **rent** as a function of $ x $:
     \[
     \text{Rent} = 400 + 8x
     \]
     where $ x $ can be positive or negative, representing increases or decreases of $8.

2. **Occupancy Equation:**
   - The starting occupancy is 120 units when the rent is $400.
   - For each $8 increase, one unit becomes vacant. For each $8 decrease, one additional unit is occupied.
   - Thus, the **number of occupied units** as a function of $ x $ is:
     \[
     \text{Occupied units} = 120 - x
     \]

3. **Revenue Function:**
   - Revenue $ R $ is calculated as the product of rent and occupied units:
     \[
     R(x) = (\text{Rent}) \times (\text{Occupied units}) = (400 + 8x)(120 - x)
     \]

4. **Expand the Revenue Function:**
   \[
   R(x) = (400 + 8x)(120 - x)
   \]
   Expanding this expression:
   \[
   R(x) = 400 \cdot 120 - 400x + 8x \cdot 120 - 8x^2
   \]
   \[
   R(x) = 48000 + 960x - 400x - 8x^2
   \]
   \[
   R(x) = 48000 + 560x - 8x^2
   \]

   So, the revenue function is:
   \[
   R(x) = -8x^2 + 560x + 48000
   \]

5. **Maximizing the Revenue:**
   - This is a quadratic function of the form $ R(x) = -8x^2 + 560x + 48000 $, which opens downwards (since the coefficient of $ x^2 $ is negative), so the maximum revenue occurs at the vertex.
   - The vertex $ x $ of a quadratic $ ax^2 + bx + c $ is given by $ x = -\frac{b}{2a} $.
   - Here, $ a = -8 $ and $ b = 560 $:
     \[
     x = -\frac{560}{2 \cdot -8} = \frac{560}{16} = 35
     \]

6. **Calculate the Optimal Rent:**
   - Substitute $ x = 35 $ back into the rent formula:
     \[
     \text{Optimal Rent} = 400 + 8 \cdot 35 = 400 + 280 = 680
     \]

### Conclusion:
The manager should charge **$680** per month to maximize revenue.

Would you like a detailed explanation on any specific part of this solution, or do you have further questions?

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Here are 5 related questions you could explore:

1. How would the optimal rent change if the rent increase or decrease was $10 instead of $8?
2. What happens to revenue if the rent is set to $500? How does it compare to the maximum revenue?
3. How would a different starting occupancy level affect the optimal rent?
4. How would the function change if each $8 increase affected 2 units instead of 1?
5. Can this method be applied to other real-estate problems with similar conditions?

**Tip:** In quadratic functions, always check if the parabola opens upwards or downwards to determine if you're looking for a minimum or maximum at the vertex.

The manager of a large apartment complex knows from experience that 120 units will be occupied if the rent is 400 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Learn how to calculate the optimal rent to maximize revenue for a large apartment complex with 120 units at $400 rent per month. This problem involves understanding revenue functions, quadratic equations, and the vertex formula. Suitable for high school students in grades 10-12.

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