The total revenue for Jane's Vacation Rentals is given as the function R(x)=400x−0.2x^2
, where x is the number of apartments filled.  What number of apartments filled produces the maximum revenue?

To find the number of apartments filled that produces the maximum revenue, we need to determine the value of \( x \) that maximizes the revenue function \( R(x) = 400x - 0.2x^2 \).

### Step 1: Identify the revenue function
The given revenue function is:
\[
R(x) = 400x - 0.2x^2
\]

### Step 2: Find the vertex of the parabola
Since this is a quadratic function of the form \( R(x) = ax^2 + bx + c \), the maximum (or minimum) value occurs at the vertex. The formula to find the \( x \)-coordinate of the vertex is:

\[
x_{\text{max}} = \frac{-b}{2a}
\]

Here, \( a = -0.2 \) and \( b = 400 \).

\[
x_{\text{max}} = \frac{-400}{2(-0.2)} = \frac{-400}{-0.4} = 1000
\]

### Step 3: Interpret the result
The maximum revenue is achieved when 1000 apartments are filled.

Thus, **1000 apartments filled** produces the maximum revenue.

Would you like more details or have any questions?

### Relative Questions:
1. What is the maximum revenue generated when 1000 apartments are filled?
2. How would the revenue function change if the price per apartment increases?
3. What would happen to the maximum revenue if the coefficient of \( x^2 \) was different?
4. Can you explain how to derive the vertex formula for a quadratic function?
5. What is the revenue if only 500 apartments are filled?

**Tip:** The vertex formula \(\frac{-b}{2a}\) is crucial in maximizing or minimizing a quadratic function, and it always points to the extremum (maximum or minimum) of the function.

Discover how to find the number of apartments filled that produces maximum revenue for Jane's Vacation Rentals. The revenue function R(x) = 400x - 0.2x^2 is analyzed step-by-step to determine the optimal number of apartments. Suitable for advanced high school students.

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