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The question on the left page asks you to analyze two models for predicting the annual exposure to particulate matter (PM) across Europe, using the given formula \( E = 60.91 \cdot 0.91^x + 13.8 \) and data from a graph. Here’s a breakdown of the steps you should take:

1. **Identify a Factor for Bias**: Consider factors like changing pollution sources, advancements in technology, or policy changes which could impact PM levels.

2. **Predict PM Levels for 2030**: The formula \( E = 60.91 \cdot 0.91^x + 13.8 \) is provided, where \( x \) is the number of years after 1990. For 2030, \( x = 40 \), so substitute \( x = 40 \) into the equation.

3. **Evaluate Accuracy**: Discuss the reliability of the model, noting if the decline in PM levels aligns with historical trends and if external factors (e.g., policy changes) could alter predictions.

4. **Provide a Recommendation**: Use the predicted value and analysis of model accuracy to recommend the best model for predicting PM levels, with reasoning.

Would you like a detailed calculation for predicting the 2030 PM level or help with any specific part of this analysis?

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Here are some related questions to deepen understanding:

1. How does the exponential model compare to a linear model in predicting future values?
2. What other environmental factors could influence particulate matter levels beyond 2030?
3. How does changing the base of the exponential function affect predictions in similar models?
4. Could sudden technological advancements significantly alter this type of environmental model?
5. Why is it crucial to consider external factors, such as policy changes, when using historical models for future predictions?

**Tip**: When working with exponential models, always check if the base value indicates growth (base > 1) or decay (base < 1). This helps in interpreting the trend direction directly.

You are a statistical advisor on a government panel that wants to reduce annual exposure to particulate matter in Europe to a level below 13 µg/m³ by 2030. Analyze the two models to make a prediction of annual exposure to particulate matter in 2030. In your answer, you should: identify a factor from the data to be considered when making your prediction, predict the level of annual exposure to particulate matter in 2030 using models from both graphs, comment on the accuracy of your predictions, and advise the government which is a better model to make predictions and justify your answer.

Learn how to analyze and compare two mathematical models to predict the annual exposure to particulate matter (PM) in Europe by 2030. This problem involves using an exponential decay function, interpreting graphs, and assessing model accuracy. Ideal for Grades 10-12, this question offers insights into environmental forecasting with math.

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