Questions: Find the maximum revenue for the revenue function R(x) = 170x - 0.80x^2. (Round your answer to the nearest cent.)

Transcript text: Find the maximum revenue for the revenue function $R(x)=170 x-0.80 x^{2}$. (Round your answer to the nearest cent.) \$ $\square$ Submit Answer

Solution

Solution Steps

To find the maximum revenue for the given revenue function $ R(x) = 170x - 0.80x^2 $, we need to determine the vertex of the parabola represented by this quadratic function. The vertex form of a quadratic function $ ax^2 + bx + c $ gives the x-coordinate of the vertex as $ x = -\frac{b}{2a} $. Once we find this x-coordinate, we can substitute it back into the revenue function to find the maximum revenue.

Step 1: Identify the Revenue Function

The revenue function is given by

\[ R(x) = 170x - 0.80x^2. \]

Step 2: Find the Vertex

To find the maximum revenue, we need to determine the x-coordinate of the vertex of the parabola. The formula for the x-coordinate of the vertex is

\[ x = -\frac{b}{2a}, \]

where $ a = -0.80 $ and $ b = 170 $. Substituting these values, we get

\[ x = -\frac{170}{2 \times -0.80} = 106.25. \]

Step 3: Calculate Maximum Revenue

Now, we substitute $ x = 106.25 $ back into the revenue function to find the maximum revenue:

\[ R(106.25) = 170(106.25) - 0.80(106.25)^2. \]

Calculating this gives