Problem 29 - Algebra Form and Function

Question

Methane is a greenhouse gas implicated as a contributor to global warming. Answer based on the table of values of \(Q=w(t),\) the atmospheric methane level in parts per billion (ppb) \(t\) years after \(1980 .^{5}\) $$\begin{aligned}&\text { Table } 4.15\\\&\begin{array}{c|c|c|c|c|c|c} \hline t & 0 & 5 & 10 & 15 & 20 & 25 \\\\\hline Q & 1575 & 1660 & 1715 & 1750 & 1770 & 1775 \\ \hline\end{array}\end{aligned}$$ Evaluate $$\frac{w(10)-w(5)}{10-5}$$ and say what it tells you about atmospheric methane levels.

Step-by-Step Solution

Answer

Answer: The expression represents the average rate of change of atmospheric methane levels between 1985 and 1990. Using the given data, the value is 11 ppb, meaning that on average, methane levels increased by 11 ppb each year during that time period.

Step 1: Identify the values of w(t) from the table

From the given table, we can see the atmospheric methane levels for the years in which \(t=5\) and \(t=10\). These values are: $$w(5) = 1660\,\text{ppb}, \hspace{1cm} w(10) = 1715\,\text{ppb}$$

Step 2: Evaluate the expression

Now, we will plug these values into the expression we want to evaluate: $$\frac{w(10)-w(5)}{10-5} = \frac{1715 - 1660}{10 - 5} = \frac{55}{5}$$ Calculating the expression: $$\frac{55}{5} = 11$$

Step 3: Interpret the result

The obtained value of \(11\) represents the average rate of change of atmospheric methane levels between 1985 and 1990. This means that, on average, the methane levels increased by \(11\,\text{ppb}\) each year during that time period. This increase in atmospheric methane levels can be considered a contributor to global warming.

Key Concepts

Greenhouse Gases

Greenhouse gases are a collection of gaseous compounds in the Earth's atmosphere that are capable of trapping heat. This heat trapping occurs because these gases absorb solar radiation and re-emit it in all directions, including back towards the Earth's surface. This process, known as the greenhouse effect, is vital for keeping the planet warm enough to sustain life.
However, human activities have significantly increased the concentrations of these gases, enhancing the natural greenhouse effect, which leads to global warming. Here are some examples of common greenhouse gases:

• Carbon dioxide (CO₂)
• Methane (CH₄)
• Nitrous oxide (N₂O)
• Water vapor (H₂O)
• Fluorinated gases

Among these, methane is particularly potent, having a greater heat-trapping ability than carbon dioxide over a shorter period of time. Addressing the emissions of these gases is essential to moderating the effects of global climate change.

Atmospheric Methane

Methane, a simple hydrocarbon with the chemical formula CH₄, is a colorless, odorless gas that is a significant greenhouse gas. Most atmospheric methane is produced through human-related activities such as agriculture, fossil fuel extraction, and waste management. Natural processes, such as wetlands decomposition, also release methane.
When we evaluate the rate of change in atmospheric methane from the data, like in the exercise, it gives us insight into how fast the concentration is rising or falling. For example, a calculated average rate of change of 11 ppb per year, as derived from the exercise, indicates a consistent increase, highlighting the growing concentration of methane in the atmosphere.
This rising level of methane is a concern due to its efficiency in trapping heat and its significant role in contributing to global warming. Therefore, monitoring and managing methane emissions is crucial for environmental protection.

Global Warming

Global warming refers to the long-term increase in Earth's average surface temperature due to human activities, especially the emission of greenhouse gases. The Earth's climate system is being influenced by these increases, leading to widespread consequences such as:

• Rising sea levels
• More frequent and severe weather events
• Loss of sea ice and glaciers
• Changes in precipitation patterns
• Disruption to ecosystems and biodiversity

Understanding the role of gases like methane in global warming is key. Although carbon dioxide is more abundant, methane's potential to trap heat is much higher in the short term. The trends in methane increase, as calculated in the exercise, can serve as indicators of the ongoing impact of human activity on our climate.
Addressing global warming requires reducing greenhouse gas emissions and transitioning to renewable energy sources, improving energy efficiency, and protecting natural carbon sinks. Each action we take can contribute to mitigating the adverse effects of global warming.