Problem 58 - Algebra A Combined Function

Question

Methane is a gas produced by landfills, natural gas systems, and coal mining that contributes to the greenhouse effect and global warming. Projected methane emissions in the United States can be modeled by the quadratic function $$ f(x)=-0.072 x^{2}+1.93 x+173.9 $$ where \(f(x)\) is the amount of methane produced in millions of metric tons and \(x\) is the number of years after 2000. (Source: Based on data from the U.S. Environmental Protection Agency, \(2000-2020)\) a. According to this model, what will U.S. emissions of methane be in \(2018 ?\) b. Will this function have a maximum or a minimum? How can you tell? C. In what year will methane emissions in the United States be at their maximum or minimum? Round to the nearest whole year. d. What is the level of methane emissions for that year? (Use your rounded answer from part c.)

Step-by-Step Solution

Answer

a) 185.3 million metric tons in 2018; b) Maximum (a < 0); c) Year 2013; d) 186.8 million metric tons.

Step 1: Identify the Value of x for 2018

To find the value of \( x \) for the year 2018, determine how many years after 2000 it is. This is simply \( 2018 - 2000 = 18 \). Thus, \( x = 18 \).

Step 2: Calculate Methane Emissions for 2018

Substitute \( x = 18 \) into the quadratic function. Compute \( f(18) = -0.072(18)^2 + 1.93(18) + 173.9 \). Evaluating, calculate: \( f(18) = -0.072 \times 324 + 34.74 + 173.9 = -23.328 + 34.74 + 173.9 = 185.312 \). Thus, the methane emissions for 2018 will be approximately \( 185.3 \) million metric tons.

Step 3: Identify Maximum or Minimum of the Function

A quadratic function in the form \( f(x) = ax^2 + bx + c \) has a maximum point if \( a < 0 \) and a minimum point if \( a > 0 \). In this quadratic function, \( a = -0.072 \), which is less than zero. Therefore, the function has a maximum.

Step 4: Determine the Year of Maximum Methane Emissions

The maximum of a quadratic function \( f(x) = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). For the given function, \( a = -0.072 \) and \( b = 1.93 \). Thus, \( x = -\frac{1.93}{2(-0.072)} = \frac{1.93}{0.144} \approx 13.4 \). Rounding to the nearest whole number, \( x = 13 \). Since \( x \) represents the number of years after 2000, the year of maximum emissions is \( 2000 + 13 = 2013 \).

Step 5: Calculate Methane Emissions for the Year 2013

Use the rounded value of \( x = 13 \) and substitute it into the function to find \( f(13) = -0.072(13)^2 + 1.93(13) + 173.9 \). Calculating, \( f(13) = -0.072 \times 169 + 25.09 + 173.9 = -12.168 + 25.09 + 173.9 = 186.822 \). Thus, the maximum level of methane emissions is approximately \( 186.8 \) million metric tons in 2013.

Key Concepts

Methane Emissions

Methane emissions are a significant concern when discussing environmental impacts. Methane, a potent greenhouse gas, is released from various sources including landfills, natural gas systems, and coal mining. When methane enters the atmosphere, it effectively traps heat, making it far more potent than carbon dioxide over short periods.
Methane emissions can be represented mathematically through models, often using quadratic functions, to predict future emission levels. The emissions discussed here are modeled as a quadratic function: \[ f(x) = -0.072x^2 + 1.93x + 173.9 \]where \( f(x) \) represents the methane emissions in millions of metric tons, and \( x \) is the number of years after the year 2000. This kind of modeling helps stakeholders understand and anticipate changes in methane levels over time, providing data necessary to enact policies aimed at emission reductions.
Understanding how to interpret these functions is crucial. By substituting different values for \( x \), you can see the projected methane emissions for specific years, which can directly inform environmental planning.

Greenhouse Effect

The greenhouse effect is a natural process where certain gases in Earth's atmosphere capture heat, preventing it from escaping back into space. While a natural greenhouse effect is essential for maintaining the livability of our planet by keeping surface temperatures comfortable, an enhancement by excess greenhouse gases can lead to global warming.
Methane is one of these greenhouse gases, and although it exists in smaller quantities compared to carbon dioxide, its heat-trapping capability is significantly more efficient. This means methane emissions have a disproportionately large impact per molecule.
This model allows scientists to study how future methane emissions could exacerbate the greenhouse effect. The more accurately these functions reflect projected emissions, the better we can anticipate the resulting changes in global temperature and address strategies to curb emissions.

Maximum and Minimum Values

In the context of quadratic functions, understanding maximum and minimum values is crucial for interpreting environmental data, like methane emissions. Quadratics are represented by the form \( ax^2 + bx + c \).

• If \( a < 0 \), the parabola opens downwards, indicating the function has a maximum value.
• If \( a > 0 \), the parabola opens upwards, indicating the function has a minimum value.

For our methane emission function, since \( a = -0.072 \), which is less than zero, the function reaches a maximum point. This maximum offers insights into when emissions peak, guiding regulations and mitigation strategies.
The maximum value of a quadratic function is calculated at \( x = -\frac{b}{2a} \). In this case, when substitutions are made, it results in a maximum year for emissions. Determining when these emissions reach their highest helps us to target specific periods for intensive environmental management and action.

Environmental Modeling

Environmental modeling involves using mathematical functions to simulate and analyze environmental processes and phenomena. It is a powerful tool for predicting variables such as methane emissions, which help scientists understand potential future changes and adapt policies accordingly.
By using quadratic functions like the one given, we can project the quantity of methane emissions across different years and create strategies based on this data.
Experts rely on these models to:

• Visualize changes over time, such as in the reduction or increase of methane emissions.
• Inform policymakers of critical periods for emission reductions.
• Enhance public understanding of the impact of methane on climate change.

Environmental modeling is pivotal in addressing human impacts on climate, facilitating both reactive and proactive measures to tackle challenges associated with the greenhouse effect and global warming. In summary, effectively leveraging environmental models ensures a comprehensive approach to mitigating potential adverse impacts on our planet.