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To find the predicted average rate of change from year 1 to year 8 for the given data modeled by a quartic function, we first need to establish the mathematical model for the data.

Given:
- \(x\) values: 1, 2, 3, 4, 5, 6
- \(h(x)\) values: 7.6, 5.9, 5.4, 4.8, 4.5, 4.7

The steps are as follows:

1. **Fit a Quartic Model:** Use regression to find a quartic polynomial \( h(x) = ax^4 + bx^3 + cx^2 + dx + e \) that approximates the given data points.

2. **Evaluate \(h(x)\) for \(x = 1\) and \(x = 8\)** using the quartic model.

3. **Calculate the Average Rate of Change:**
   \[
   \text{Average rate of change} = \frac{h(8) - h(1)}{8 - 1}
   \]

Let me perform these calculations for you. I'll start with fitting the quartic model.The predicted values using the quartic model are:
- \( h(1) \approx 7.59 \)
- \( h(8) \approx 13.41 \)

The average rate of change from year 1 to year 8 is:
\[
\frac{13.41 - 7.59}{8 - 1} \approx 0.831
\]

Thus, the predicted average rate of change is approximately **0.832**.

Would you like more details or have any questions?

### Related Questions:
1. How is a quartic model different from linear or quadratic models?
2. Why might a quartic model be suitable for this data set?
3. What are the implications of using higher-degree polynomials in modeling?
4. How can we verify the accuracy of a model?
5. What other methods could be used to estimate the rate of change?

### Tip:
When fitting polynomial models, be cautious of overfitting, especially with high-degree polynomials, as they can produce misleading trends outside the given data range.

The table gives data on lead emissions into the environment in millions of metric tons for certain years. If the data is known to be modeled by a quartic model, what is the predicted average rate of change from year 1 to year 8?

Explore how to predict the average rate of change in lead emissions using a quartic polynomial model. This example involves polynomial regression and step-by-step calculations to estimate emissions between year 1 and year 8, providing insights into modeling environmental data.

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