Find the maximum revenue for the revenue function $$ R(x)=385 x-0.6 x^{2} $$. (Round your answer to the nearest cent.) $$ R=\$ \square $$

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To find the maximum revenue for the function $$ R(x) = 385x - 0.6x^2 $$, we can follow these steps:

1. **Identify the Function Type**:
   - The given function is a quadratic equation of the form $$ R(x) = ax^2 + bx + c $$, where $$ a = -0.6 $$ and $$ b = 385 $$.
   - Since $$ a $$ is negative, the graph is a downward-opening parabola, and the vertex represents the maximum point.

2. **Find the Vertex**:
   - The x-coordinate of the vertex for a quadratic function is given by:
     $$
     x = \frac{-b}{2a}
     $$
   - Plugging in the values:
     $$
     x = \frac{-385}{2(-0.6)} = \frac{385}{1.2} \approx 320.83
     $$

3. **Calculate the Maximum Revenue**:
   - Substitute $$ x \approx 320.83 $$ back into the revenue function:
     $$
     R(320.83) = 385(320.83) - 0.6(320.83)^2
     $$
   - Alternatively, using the vertex formula for maximum value:
     $$
     R_{\text{max}} = \frac{b^2}{4|a|}
     $$
     $$
     R_{\text{max}} = \frac{385^2}{4 \times 0.6} = \frac{148225}{2.4} \approx 61760.42
     $$

4. **Final Answer**:
   - Rounding to the nearest cent, the maximum revenue is:

$$
R = \$61,760.42
$$

**Answer:**  
$$ R = \$61,\!760.42 $$
The maximum revenue is \$61,760.42.

Find the maximum revenue for the revenue function \( R(x)=385 x-0.6 x^{2} \). (Round your answer to the nearest cent.) \( R=\$ \square \)

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