Let's analyze the given linear function:

$T = 0.02t + 8.50$

where:

• $T$ represents the temperature in °C.
• $t$ represents the number of years since 1900.

(a) Interpreting the slope and T-intercept

The general form of a linear equation is:
$y = mx + b$
where $m$ is the slope and $b$ is the y-intercept (or T-intercept in this case).

• Slope: The coefficient of $t$ is 0.02. This means that the average surface temperature of the planet is increasing at a rate of 0.02°C per year.
• T-intercept: The value of $T$ when $t = 0$ is 8.50. This represents the average surface temperature in the year 1900.

(b) Predicting the temperature in 2100

To find the temperature in the year 2100, substitute $t = 2100 - 1900 = 200$ into the equation:

$T = 0.02(200) + 8.50$

$T = 4 + 8.50 = 12.50°C$

So, the predicted temperature in 2100 is 12.50°C.

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Final Answers:

(a)

• Slope = 0.02, meaning the temperature is increasing at a rate of 0.02°C per year.
• T-intercept = 8.50, meaning the temperature in 1900 was 8.50°C.

(b)

• The predicted temperature in 2100 is 12.50°C.

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Would you like a deeper explanation of any part?

Here are five related questions to explore:

1. How would the equation change if the temperature were rising at 0.03°C per year instead?
2. What year will the average temperature reach 15°C according to this model?
3. What assumptions does this linear model make about temperature change?
4. How does this compare to actual recorded temperature trends?
5. Could the relationship between temperature and time be nonlinear? If so, why?

Tip: Always check whether a linear model is reasonable for long-term predictions, as climate trends can be influenced by many complex factors! 🌍