To solve this problem, we'll follow the steps carefully.

(a) Create a linear function

We want to model the data using a linear equation in the form:

$y(x) = mx + b$

where:

• $y(x)$ is the millions of metric tons of carbon dioxide emissions.
• $x$ is the number of years past 2010, so for example, in 2012, $x = 2$, in 2014, $x = 4$, and so on.
• $m$ is the slope, representing the rate of change in CO2 emissions per year.
• $b$ is the y-intercept, representing the emissions in 2010.

Step 1: Convert years into "years past 2010"

We will use the following data points:

Year	$x$ (Years since 2010)	CO2 Emissions (y)
2010	0	338.5
2012	2	363.5
2014	4	398.1
2016	6	425.8
2018	8	454.1
2020	10	497.4
2022	12	556.2
2024	14	590.9
2026	16	629.7
2028	18	662.1
2030	20	709.1
2032	22	741.7

Step 2: Calculate the slope $m$

To calculate the slope $m$, we use the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using two points, say (0, 338.5) and (22, 741.7), we can compute the slope.

Let me calculate it.The slope $m$ of the linear model is approximately $18.33$. This means that, on average, the carbon dioxide emissions increase by about 18.33 million metric tons per year.

Step 3: Find the y-intercept $b$

Now that we know the slope, we can use the point $(0, 338.5)$ (representing the year 2010) to solve for the y-intercept $b$. The formula is:

$b = y_1 - m \cdot x_1$

Since $x_1 = 0$, we have $b = 338.5$.

Thus, the linear model is:

$y(x) = 18.33x + 338.5$

(b) Find the model's estimate for the 2022 data point

For 2022, $x = 12$. We will substitute $x = 12$ into the linear equation to estimate the CO2 emissions in that year:

$y(12) = 18.33 \cdot 12 + 338.5$

Let me calculate that.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the estimate for 2022 by plugging $x = 12$ into the equation:

$y(12) = 18.33 \times 12 + 338.5 = 558.46$

So, the estimated CO2 emissions for 2022 are approximately 558.46 million metric tons.

(c) Find the slope and its interpretation

The slope of the linear model is $18.33$, which means that for each year since 2010, carbon dioxide emissions are expected to increase by approximately 18.33 million metric tons.

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Would you like to discuss any further details or calculations? Here are five related questions to expand your understanding:

1. How would this model differ if we used a quadratic function instead of a linear one?
2. Can the model be used to predict emissions beyond 2032, and what limitations might arise?
3. How does the increase in CO2 emissions impact global climate projections?
4. What other factors might affect the linear growth of CO2 emissions in the real world?
5. How would a decrease in emissions alter the equation and slope?

Tip: The slope in a linear model represents the rate of change, which can help forecast future trends based on past data.