https://file.aiquickdraw.com/mathbot/file/1728441823049qft0cm6a.jpg

Let's work through the "Prep Activity 4.5.2" problems in the image.

### Part (a): Explain why it’s reasonable to find a linear model for this trend.
The graph shows annual mean CO₂ levels from 2006 to 2022. The data appears to increase steadily year by year, forming an upward trend with no sharp fluctuations or breaks. Since the increase in CO₂ levels follows a relatively straight line, it suggests that the rate of change is consistent over time. Therefore, it is reasonable to approximate the data using a linear model, which assumes that CO₂ levels increase by a fixed amount each year.

### Part (b): Sketch a line on the graph that fits the general trend, and use your line to find a formula for an appropriate linear model.
You can sketch a line that roughly connects the dots of the data points on the graph. We can estimate the slope and y-intercept by looking at two points from the graph.

Using two points (for example, (2006, 381 ppm) and (2022, 417 ppm)):
1. Calculate the slope (rate of change) using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{417 - 381}{2022 - 2006} = \frac{36}{16} = 2.25 \, \text{ppm per year}
\]

2. Use the point-slope form to write the equation. We'll use the point (2006, 381):

\[
y - 381 = 2.25(x - 2006)
\]

Simplify to get the equation of the line:

\[
y = 2.25x - 4521 + 381 \quad \Rightarrow \quad y = 2.25x - 4140
\]

Thus, the linear model is approximately:

\[
y = 2.25x - 4140
\]

### Part (c): Using your model, predict how much CO₂ there will be in the atmosphere at Mauna Loa in 2050. How much confidence do you have in your prediction? Explain your answer.
Using the linear model \(y = 2.25x - 4140\), substitute \(x = 2050\) to predict the CO₂ level in 2050:

\[
y = 2.25(2050) - 4140 = 4612.5 - 4140 = 472.5 \, \text{ppm}
\]

So, the predicted CO₂ level in 2050 is approximately 472.5 ppm.

#### Confidence in the prediction:
This model assumes that the rate of increase in CO₂ will continue in a linear fashion. While the data so far follows a linear trend, environmental changes, policy actions, or technological advancements may affect future CO₂ levels. Thus, while the prediction is mathematically sound, real-world confidence in the prediction would depend on future developments that could alter the rate of CO₂ emissions.

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Do you have any questions, or would you like more details on any part?

### Follow-up questions:
1. How would a nonlinear model differ from a linear model for predicting CO₂ levels?
2. What environmental factors could influence the accuracy of this prediction?
3. How would reducing CO₂ emissions globally impact future CO₂ levels compared to this model?
4. Can you describe how confidence intervals could be used to express the uncertainty in your prediction?
5. What other atmospheric measurements might be useful in conjunction with CO₂ levels to model climate change?

### Tip:
When creating a model based on data, always consider the potential for external factors to influence future trends—especially for long-term predictions!

Prep Activity 4.5.2: The graph shows annual mean CO₂ levels from 2006 to 2022, and the user is tasked with explaining why it is reasonable to find a linear model for this trend, sketching a line to find a formula, and predicting future CO₂ levels in 2050.

Learn how to model the increase in CO₂ levels using data from the Mauna Loa Observatory between 2006 and 2022. This activity involves finding a linear model, calculating slope, and predicting future CO₂ concentrations in 2050, suitable for high school students (Grades 9-12).

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