Let's break down the linear equation provided:

The equation given is: $y = 0.031x + 57.2$

Where:

• $y$ represents the average temperature in degrees Fahrenheit,
• $x$ represents the number of years since 1975.

1. Slope:

The slope is the coefficient of $x$, which is $0.031$. This means the temperature increases by $0.031$ degrees Fahrenheit each year.

2. Y-intercept:

The y-intercept is the constant term, which is $57.2$. This represents the average temperature in 1975, since when $x = 0$ (the base year of 1975), $y = 57.2$.

3. Predicting the average temperature in 2021:

To predict the temperature for the year 2021, we calculate $x$ for 2021. Since $x$ represents the number of years since 1975: $x = 2021 - 1975 = 46$

Now, substitute $x = 46$ into the equation: $y = 0.031(46) + 57.2$ $y = 1.426 + 57.2 = 58.626$

Thus, the predicted average surface temperature in 2021 is approximately 58.63°F.

4. Predicting the average temperature for 2100:

To predict the temperature for 2100, we calculate $x$ for 2100: $x = 2100 - 1975 = 125$

Now, substitute $x = 125$ into the equation: $y = 0.031(125) + 57.2$ $y = 3.875 + 57.2 = 61.075$

Thus, the predicted average surface temperature in the year 2100 is approximately 61.08°F.

Would you like further details on any of the steps? Let me know if you have any questions!

Here are some related questions for deeper understanding:

1. How does the slope of the line influence the temperature trend over time?
2. What would happen to the temperature if the slope were increased to 0.05?
3. How could you adjust the equation if the starting year were 1980 instead of 1975?
4. How might this linear model change if there were significant environmental changes after 2021?
5. What other types of equations could be used to model temperature change over time (e.g., exponential, quadratic)?

Tip:

When using a linear model, ensure the units for time (years) and temperature (degrees Fahrenheit) are consistent throughout your calculation to avoid errors.