To find the quadratic regression equation for this data set, we can follow these steps:

1. Identify the data points: From the table, the data points are: $(10, 417), (11.5, 536), (14.75, 698), (20.75, 643), (24, 417), (25.75, 247)$

2. Quadratic regression equation: A quadratic regression model has the form: $y = ax^2 + bx + c$ where $a$, $b$, and $c$ are the coefficients we need to determine.

3. Calculate coefficients: By performing a quadratic regression analysis on the data points, we find values for $a$, $b$, and $c$ (rounded to the nearest hundredth).

4. Use the equation to predict profit: Once we have the quadratic equation, we can substitute $x = 22.5$ to find the profit at this price level.

Let me calculate these coefficients for you.The quadratic regression equation for this data, with rounded coefficients, is:

$y = -6.36x^2 + 216.63x - 1113$

Using this equation, the predicted profit for a selling price of $x = 22.5$ dollars is approximately:

$\text{Profit} = 539 \text{ dollars}$

Would you like further details on the calculation steps, or do you have any questions?

Here are five related questions for further exploration:

1. How would the equation change if additional data points were added?
2. Can we use this equation to predict profit for prices outside the given range?
3. What does the negative coefficient of $x^2$ imply about the profit pattern?
4. How accurate is quadratic regression compared to linear regression for this data?
5. What other types of functions could model this data, and how would they compare?

Tip: When dealing with regression problems, always check the residuals to ensure the model fits well across all data points.