The following table gives the millions of metric tons of carbon dioxide emissions in a certain country for selected years from 2010 and projected to 2032.
Year	2010	2012	2014	2016	2018	2020
CO2 Emissions	338.5	363.5	398.1	425.8	454.1	497.4
Year	2022	2024	2026	2028	2030	2032
CO2 Emissions	556.2	590.9	629.7	662.1	709.1	741.7
(a) Create a linear function that models these data, with x as the number of years past 2010 and y as the millions of metric tons of carbon dioxide emissions. (Round all numerical values to two decimal places.)
y(x) = 
 

(b) Find the model's estimate for the 2022 data point. (Round your answer to two decimal places.)
 million metric tons

(c) Find the slope of the linear model. (Round your answer to two decimal places.)


Interpret the slope of the linear model.
For each year since 
---Select---
 , carbon dioxide emissions in the U.S. are expected to change by 
 million metric tons.

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**.

---

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.

The following table gives the millions of metric tons of carbon dioxide emissions in a certain country for selected years from 2010 and projected to 2032.
Year 2010 2012 2014 2016 2018 2020
CO2 Emissions 338.5 363.5 398.1 425.8 454.1 497.4
Year 2022 2024 2026 2028 2030 2032
CO2 Emissions 556.2 590.9 629.7 662.1 709.1 741.7
(a) Create a linear function that models these data, with x as the number of years past 2010 and y as the millions of metric tons of carbon dioxide emissions. (Round all numerical values to two decimal places.) y(x) = 
(b) Find the model's estimate for the 2022 data point. (Round your answer to two decimal places.)
(c) Find the slope of the linear model. (Round your answer to two decimal places.) Interpret the slope of the linear model. For each year since , carbon dioxide emissions in the U.S. are expected to change by million metric tons.

Learn how to create a linear function to model CO2 emissions over time using real data from 2010 to 2032. Find the slope, intercept, and estimate future emissions using algebraic techniques. This problem is ideal for high school students studying linear functions and slope-intercept form.

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