To determine the $R^2$ value for the best fit of the given data using a linear, quadratic, or exponential model, we can follow these steps:

1. Input the data points:

   • $x$ values: $[8, 24, 11, 19, 17, 9, 14, 23, 22]$
   • $y$ values: $[1.72, 11.99, 2.91, 7.70, 6.12, 2.13, 4.46, 11.07, 10.15]$

2. Fit the data to different models:

   • Linear model: $y = ax + b$
   • Quadratic model: $y = ax^2 + bx + c$
   • Exponential model: $y = ae^{bx}$

3. Calculate the $R^2$ value for each model and compare them to determine the best fit.

Let me perform these calculations for you.The $R^2$ values for the best fit of the data using different models are:

• Linear model: $R^2 = 0.9834$
• Quadratic model: $R^2 = 0.9998$
• Exponential model: $R^2 = 0.9927$

The quadratic model provides the best fit with an $R^2$ value of 0.9998.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How does the $R^2$ value change with different sets of data?
2. What does the $R^2$ value indicate about the model's fit?
3. How can you interpret the coefficients in the quadratic model?
4. Why might the quadratic model fit better than the linear or exponential models?
5. What are some limitations of using the $R^2$ value as a measure of fit?

Tip: Always consider the context of your data when choosing a model; a higher $R^2$ value does not always mean the model is appropriate for prediction or interpretation.