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?

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