To find the maximum revenue for the given revenue function $R(x) = 392x - 0.9x^2$, we need to determine the value of $x$ that maximizes $R(x)$. This can be done by finding the vertex of the parabola, since the function is quadratic.

Steps:

1. The formula for the vertex of a quadratic function $ax^2 + bx + c$ is $x = \frac{-b}{2a}$, where:

   • $a = -0.9$
   • $b = 392$

2. Apply the formula for $x$: $x = \frac{-392}{2(-0.9)} = \frac{-392}{-1.8}$

Let me calculate that for you.The value of $x$ that maximizes the revenue is approximately $217.78$.

Next, to find the maximum revenue, substitute $x = 217.78$ into the revenue function $R(x) = 392x - 0.9x^2$.

Let me calculate that for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through calculating the maximum revenue manually, or do you have any other questions?

Here are 5 related questions for further exploration:

1. How do you determine the vertex of any quadratic function?
2. What is the significance of the second derivative in identifying maxima or minima?
3. How can you interpret the result of the maximum revenue in a real-world business scenario?
4. How does the coefficient of $x^2$ affect the shape of a parabola?
5. How can you verify if a quadratic function opens upwards or downwards?

Tip: The coefficient of $x^2$ determines whether a parabola opens upwards (positive) or downwards (negative), impacting the location of maxima or minima.