To solve this problem, we need to create a linear function that models the carbon dioxide emissions data. We'll proceed step-by-step to find the correct linear model, verify its accuracy, and answer the related questions.

(a) Create a Linear Function

Let's assume the linear function has the form:

$y(x) = mx + b$

where:

• $y(x)$ represents the millions of metric tons of carbon dioxide emissions.
• $x$ is the number of years past 2010.
• $m$ is the slope (rate of change in emissions per year).
• $b$ is the y-intercept (emissions in 2010).

To find $m$ and $b$, we'll use two points from the data. Let's use the first and the last points for simplicity:

1. For the year 2010 ($x = 0$), $y = 338.5$
2. For the year 2032 ($x = 22$), $y = 742.7$

Step 1: Calculate the Slope $m$

The formula for the slope $m$ is:

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

Substitute the values:

$m = \frac{742.7 - 338.5}{22 - 0}$

Let's calculate the slope.The slope $m$ is approximately $18.37$ million metric tons per year.

Step 2: Calculate the y-intercept $b$

Now, we use the slope $m$ and one of the points to find the y-intercept $b$. Using the point for the year 2010 ($x = 0, y = 338.5$):

$y = mx + b \implies 338.5 = 18.37 \times 0 + b$

Since $x = 0$, this simplifies to:

$b = 338.5$

So, the y-intercept $b$ is $338.5$.

Linear Function

Now, we have both the slope and the y-intercept, so the linear function that models the data is:

$y(x) = 18.37x + 338.5$

(b) Find the Model's Estimate for the 2024 Data Point

To estimate the data point for the year 2024, we find $x$ as the number of years past 2010:

$x = 2024 - 2010 = 14$

Substitute $x = 14$ into the linear model:

$y(14) = 18.37 \times 14 + 338.5$

Let's calculate this value.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the steps to find the answer manually.

To find the estimate for the year 2024 using the linear function:

1. Find $x$ for the year 2024: $x = 2024 - 2010 = 14$

2. Substitute $x = 14$ into the linear model: $y(14) = 18.37 \times 14 + 338.5$

3. Calculate: $y(14) = 257.18 + 338.5 = 595.68$

So, the estimated CO₂ emissions for 2024 are approximately 595.68 million metric tons.

Would you like to proceed with more questions or need further clarification? Here are some related questions to consider:

1. What is the estimated CO₂ emission in 2030 using the same model?
2. How would the linear model change if the emission data were revised?
3. Can a non-linear model better fit the given data?
4. What are the implications of a constant increase in CO₂ emissions for climate policy?
5. How does this data compare to global CO₂ emission trends?

Tip: Understanding how to create and interpret linear models is essential for predicting future trends based on current data.