To solve this problem using a least-squares regression method for a quadratic relationship, we should focus on the given formula for $v = a (\Delta P)^b$. This can be transformed into a linear form by taking the natural logarithm of both sides:

$\ln v = \ln a + b \ln (\Delta P)$

Let $y = \ln v$, $x = \ln (\Delta P)$, $A = \ln a$, and $b = b$ (as it remains the same). Now, we have a linear form:

$y = A + b \cdot x$

Steps to Perform Least Squares Regression

1. Transform the Data:

   • Compute $x_i = \ln (\Delta P_i)$ and $y_i = \ln (v_i)$ for each data point.

2. Apply Linear Regression:

   • Use the least squares method on the transformed data to find the best-fit line, giving us values for $A$ and $b$.
   • Once $A$ is found, $a$ can be obtained as $a = e^A$.

3. Calculate Parameters:

   • Using the least squares formula, calculate $b$ (slope) and $A$ (intercept).

Let’s Perform the Calculations

I'll proceed with these calculations to obtain the parameters $a$ and $b$ that fit the data given.The parameters obtained through the least-squares regression method are as follows:

• $a \approx 0.698$
• $b \approx 0.500$

This means that the best-fit model for the data is:

$v = 0.698 \cdot (\Delta P)^{0.500}$

I have also transformed the data into $\ln(\Delta P)$ and $\ln(v)$ for each data point, which is necessary for this regression approach. Let me know if you want further details on the calculation process or the interpretation of these results.

Follow-up Questions

1. How would this model change if different units were used for $v$ and $\Delta P$?
2. Can we apply a similar method to non-power relationships?
3. How would errors or noise in data affect the accuracy of our regression parameters?
4. Why is logarithmic transformation useful in power-law regression?
5. What alternative methods exist for fitting nonlinear data?

Tip

Always verify the fit by examining residuals or plotting the model against the data to ensure it accurately represents the trend.