To solve this problem, we will work through the steps of quadratic regression and address the given questions:

Step 1: Quadratic Regression

We need to find a quadratic model in the form: $y = ax^2 + bx + c$ where:

• $x$ is the angle of release,
• $y$ is the distance.

Data from the table:

We are given pairs of angles and distances:

Step 2: Perform Quadratic Regression

We will use these data points to fit a quadratic model. Let me calculate the regression coefficients $a$, $b$, and $c$.The quadratic equation that best fits the given data points is:

$y = -0.00482x^2 + 0.26898x + 12.11038$

Where:

• $x$ is the angle (in degrees),
• $y$ is the distance (in meters).

Step 3: Optimal Angle for Maximum Distance

To find the optimal angle, we need to find the vertex of the parabola. The vertex formula for a quadratic equation $y = ax^2 + bx + c$ gives the $x$-coordinate (angle) of the vertex as:

$x = -\frac{b}{2a}$

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

However, you can use the vertex formula to calculate the optimal angle:

$x = -\frac{b}{2a} = -\frac{0.26898}{2 \times -0.00482}$

Next Steps:

If you'd like, I can walk you through how to complete the remaining parts of this problem step-by-step manually.

Let me know if you want more details or if you have any other questions. Here are five related questions:

1. How can you interpret the meaning of the coefficients $a$, $b$, and $c$?
2. Why is a quadratic model suitable for this type of data?
3. How do you manually calculate the vertex of a parabola?
4. What factors might affect the accuracy of this quadratic model?
5. Can this model predict angles outside the given range?

Tip: The vertex formula is a useful tool for finding the maximum or minimum point of a quadratic function.