The total revenue for Dante's Villas is given as the function $$ R(x)=700 x-0.5 x^{2} $$, where $$ x $$ is the number of villas rented. maximum revenue?

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To determine the **maximum revenue** for Dante's Villas given the revenue function:

$$
R(x) = 700x - 0.5x^2
$$

where $$ x $$ is the number of villas rented, follow these steps:

### 1. Identify the Type of Function
The revenue function $$ R(x) = -0.5x^2 + 700x $$ is a **quadratic function** in the standard form $$ R(x) = ax^2 + bx + c $$, where:
- $$ a = -0.5 $$
- $$ b = 700 $$
- $$ c = 0 $$

Since the coefficient of $$ x^2 $$ (**a**) is negative ($$ a = -0.5 $$), the parabola opens **downwards**, indicating that the function has a **maximum point**.

### 2. Find the Number of Villas for Maximum Revenue
The vertex of a quadratic function $$ ax^2 + bx + c $$ occurs at:

$$
x = -\frac{b}{2a}
$$

Plugging in the values:

$$
x = -\frac{700}{2 \times (-0.5)} = -\frac{700}{-1} = 700
$$

So, **700 villas** need to be rented to achieve maximum revenue.

### 3. Calculate the Maximum Revenue
Substitute $$ x = 700 $$ back into the revenue function:

$$
R(700) = 700 \times 700 - 0.5 \times (700)^2
$$
$$
R(700) = 490,000 - 0.5 \times 490,000
$$
$$
R(700) = 490,000 - 245,000 = 245,000
$$

### **Conclusion**
The **maximum revenue** Dante's Villas can achieve is **\$245,000** when **700 villas** are rented.

**Summary:**
- **Number of Villas for Maximum Revenue:** 700
- **Maximum Revenue:** \$245,000
To find the maximum revenue, set the number of villas rented to 700, which gives a maximum revenue of \$245,000.

The total revenue for Dante's Villas is given as the function \( R(x)=700 x-0.5 x^{2} \), where \( x \) is the number of villas rented. maximum revenue?

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