To address the tasks outlined in the image, I'll break them into steps:

2(a): Quadratic Regression Model for Braking Distance

We are tasked with creating a quadratic regression model for braking distance $D_B$ in terms of velocity $V$. A general quadratic model is:

$D_B = aV^2 + bV + c$

Using this equation:

• You'll need the given data (velocity and corresponding braking distances) to determine the coefficients $a$, $b$, and $c$ through regression.

Additionally, to assess the fit of the model:

• Compute the $R^2$-value, which indicates how well the quadratic model explains the variability of the braking distance.

2(b): Linear Regression Model for Braking Distance

For linear regression, the braking distance $D_B$ is modeled as:

$D_B = mV + c$

Here:

• Use the same data to determine the slope $m$ and intercept $c$.
• Compare the $R^2$-values of the quadratic and linear models. The model with a higher $R^2$ is the better fit.

2(c): Completing the Table

Using the more accurate model (based on the better $R^2$), calculate the braking distances for the velocities provided in the table:

• $25, 35, 55, 65, 75$ mph.

If you can provide the dataset or additional numerical details, I can calculate the models and complete the table for you.

Would you like me to proceed with further calculations? If you have the dataset, please provide it or describe the values.

5 Relative Questions

1. What data is used for velocity and braking distance in this scenario?
2. How is the $R^2$-value interpreted in both regression models?
3. What are the advantages of quadratic regression over linear regression in this case?
4. Can external factors (e.g., road type, tire condition) impact braking distance predictions?
5. How do you confirm whether the quadratic model aligns with physical principles of motion?

Tip

Regression models rely on accurate data, so ensure the dataset is free of anomalies and includes a wide range of velocities for reliable predictions.