Over the past​ century, global emissions of carbon dioxide from the burning of fossil fuels have risen at an average rate of about 4​% per year. If emissions continue to increase at this​ rate, about how much higher will total emissions be in 2042 than they were in​ 2010?

This problem involves exponential growth, as the carbon dioxide emissions are increasing at a constant rate of 4% per year. The formula for exponential growth is given by:

\[
P(t) = P_0 \times (1 + r)^t
\]

Where:
- \( P(t) \) is the value after \( t \) years,
- \( P_0 \) is the initial value (the amount of emissions in 2010),
- \( r \) is the growth rate (4% or 0.04),
- \( t \) is the number of years between 2010 and 2042, which is \( t = 2042 - 2010 = 32 \).

We want to find the factor by which emissions increase from 2010 to 2042, which is:

\[
\text{Increase factor} = (1 + 0.04)^{32}
\]

Now, let's calculate that.The emissions in 2042 will be approximately 3.51 times higher than they were in 2010, assuming a continuous 4% annual growth rate.

Would you like further details or have any other questions? 

Here are five related questions:
1. How does exponential growth differ from linear growth in environmental models?
2. What would happen if the growth rate of emissions were to decrease to 2% per year?
3. How can the formula for exponential growth be applied to population growth?
4. What are the potential environmental impacts of unchecked carbon dioxide emissions?
5. How might renewable energy impact the growth rate of carbon emissions?

**Tip:** Exponential growth can lead to very large numbers over time, making small percentage increases seem much larger in the long run.

Learn how to calculate the increase in carbon dioxide emissions over time using the exponential growth formula. This example problem calculates the increase in global carbon emissions from 2010 to 2042 with a continuous 4% annual growth rate. Suitable for students in Grades 10-12.

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