Problem 109 - A Graphical Approach to College Algebra

Question

A common air pollutant responsible for acid rain is sulfur dioxide \(\left(\mathrm{SO}_{2}\right) .\) Emissions of \(\mathrm{SO}_{2}\) during year \(x\) are computed by \(f(x)\) in the table. Emissions of carbon monoxide (CO) are computed by \(g(x)\) Amounts are given in millions of tons. $$\begin{array}{c|c|c|c|c|c}\boldsymbol{x} & 1970 & 1980 & 1990 & 2000 & 2010 \\\\\hline \boldsymbol{f}(\boldsymbol{x}) & 31.2 & 25.9 & 23.1 & 16.3 & 13.0 \\\\\hline \boldsymbol{g}(\boldsymbol{x}) & 204.0 &185.4 & 154.2 & 114.5 & 74.3\end{array}$$ (a) Evaluate \((f+g)(2010)\) (b) Interpret \((f+g)(x)\) (c) Make a table for \((f+g)(x)\)

Step-by-Step Solution

Answer

(f+g)(2010) is 87.3 million tons; it gives total emissions for SO2 and CO; see table for (f+g)(x).

Step 1: Evaluate \((f+g)(2010)\)

To find \((f+g)(2010)\), we need to add the values of \(f(x)\) and \(g(x)\) for the year 2010. The value of \(f(2010)\) is 13.0 million tons, and the value of \(g(2010)\) is 74.3 million tons. Thus, \((f+g)(2010) = f(2010) + g(2010) = 13.0 + 74.3 = 87.3\) million tons.

Step 2: Interpret \((f+g)(x)\)

The function \((f+g)(x)\) represents the total combined emissions of sulfur dioxide (\(SO_2\)) and carbon monoxide (CO) for a given year \(x\). It gives the sum of these two pollutants' emissions, encapsulating the total air pollution contribution by these specific substances.

Step 3: Make a table for \((f+g)(x)\)

To create a table for \((f+g)(x)\), evaluate \((f+g)(x)\) for each year by adding the corresponding \(f(x)\) and \(g(x)\) values. Here's the table:\[\begin{array}{|c|c|}\hline \boldsymbol{x} & \boldsymbol{(f+g)(x)} \ \hline 1970 & 235.2 \ 1980 & 211.3 \ 1990 & 177.3 \ 2000 & 130.8 \ 2010 & 87.3 \ \hline\end{array}\]This provides the total emissions for each specified year.

Key Concepts

Emissions Calculation

Understanding how emissions are calculated is key when dealing with environmental data. It's all about figuring out how much of a pollutant is released into the atmosphere over a certain period, which in this case is measured in millions of tons per year. The functions given, \(f(x)\) and \(g(x)\), represent emissions of two different pollutants: sulfur dioxide (\(\mathrm{SO}_2\)) and carbon monoxide (CO), respectively.
To calculate the combined emissions for any given year \(x\), we simply sum the individual emissions from each function. For example, if we need to evaluate \((f+g)(2010)\), it involves adding \(f(2010)\), which is 13.0 million tons, to \(g(2010)\), amounting to 74.3 million tons. Thus, \((f+g)(2010)\) equals 87.3 million tons.

• Emissions of \(\mathrm{SO}_2\) are represented by \(f(x)\).
• Emissions of CO are represented by \(g(x)\).
• The combined function \((f+g)(x)\) signifies total emissions for any year \(x\).

This calculation helps us grasp the extent of air pollution for specific years, giving us insight into environmental changes over time.

Table Creation

Creating a table is an effective way to organize data and results clearly. For functions like \((f+g)(x)\), tables help illustrate the trend and variability in emissions over several years. It involves evaluating the function for each year of interest and documenting those findings in a clear format.
In this particular example, we create a table for the years 1970, 1980, 1990, 2000, and 2010. By calculating \((f+g)(x)\) for each year, which means simply adding the \(f(x)\) and \(g(x)\) values for those years and listing the result.

• For 1970: \((f+g)(1970) = 31.2 + 204.0 = 235.2\) million tons
• For 1980: \((f+g)(1980) = 25.9 + 185.4 = 211.3\) million tons
• For 1990: \((f+g)(1990) = 23.1 + 154.2 = 177.3\) million tons
• For 2000: \((f+g)(2000) = 16.3 + 114.5 = 130.8\) million tons
• For 2010: \((f+g)(2010) = 13.0 + 74.3 = 87.3\) million tons

The result is a simple yet comprehensive table showing how the total emissions have changed, highlighting either improvements or declines in environmental health over decades.

Pollution Interpretation

Interpreting pollution data means understanding the broader environmental impact and what these specific numbers tell us. The function \((f+g)(x)\) provides insights into the total combined emissions of \(\mathrm{SO}_2\) and CO over time.
These interpretations help us see the progress or setbacks in reducing air pollutants. Lower values in \((f+g)(x)\) over a span of years suggest effective pollution control measures and possibly stricter regulations. Conversely, higher values may indicate increased industrial activity or less stringent policies at that time.

• Reductions in emissions over time reflect successful environmental policies.
• High emission values could relate to increased industrial output.
• Total emissions give a snapshot of air quality and potential health impacts for humans and wildlife.

Interpreting these trends is crucial for policymakers, environmentalists, and the public to make informed decisions about future actions to protect our environment and health.