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?

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.