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Let's walk through the problem.

The table provides the following data points:

| \( x \) | \( y \) |
|--------|---------|
| 0      | 10.2    |
| 5      | 2.9     |
| 10     | 8.2     |
| 15     | 16.1    |
| 20     | 17.9    |

We are tasked with finding three different models (linear, quadratic, and cubic) that fit the data and then determining which model best fits the data, as well as using the best-fit model to predict the value of \( y \) when \( x = 25 \).

### Steps:
1. **Linear Model**: A linear model has the form \( y = ax + b \).
2. **Quadratic Model**: A quadratic model has the form \( y = ax^2 + bx + c \).
3. **Cubic Model**: A cubic model has the form \( y = ax^3 + bx^2 + cx + d \).

We will use the data points and fit these three models. I will calculate the coefficients of each model and then use the best fit to predict the value of \( y \) when \( x = 25 \).

Let me perform these calculations and give you the answers.Here are the results for the models based on the dataHere are the results for the models based on the 

1. **Linear model**: \( y = 0.572x + 5.34 \)
2. **Quadratic model**: \( y = 0.059x^2 - 0.617x + 8.311 \)
3. **Cubic model**: \( y = -0.012x^3 + 0.433x^2 - 3.297x + 10.181 \)

### Best Fit Model
The cubic model generally provides the best fit for data with more variability, and in this case, it accounts for the nonlinear pattern in the data. Based on the more complex shape of the data, the **cubic model** seems to fit the data best.

### Prediction for \( x = 25 \)
Using the cubic model to predict the value of \( y \) when \( x = 25 \), we get:
\[ y(25) \approx 3.86 \]

Let me know if you'd like a more detailed explanation of any part of the solution or have any other questions!

### Questions:
1. Why is the cubic model preferred over the linear and quadratic models?
2. Can you explain how the coefficients of the cubic model influence the curve's shape?
3. How would the prediction differ if we used the linear or quadratic model?
4. Can the cubic model be used reliably for values of \( x \) far from the given data points?
5. How would the fitting process change if more data points were added?

### Tip:
When choosing a best-fit model, always consider the complexity of the data and try higher-degree polynomials if the pattern is non-linear.

There are times when we need to use a polynomial model of degree three or greater. Consider the table given below. Use your calculator to find a linear model, a quadratic model, and a cubic model. Then, determine which model best fits the data and predict the value of y when x = 25.

Learn how to fit linear, quadratic, and cubic polynomial models to data points using a polynomial regression approach. This problem covers key concepts including finding the best-fit model and predicting future values. Suitable for high school students.

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