91Ó°ÊÓ

The greenhouse gas methane lets sunlight into the atmosphere, but blocks heat from escaping the earth's atmosphere. Methane is a by-product of burning fossil fuels. In the table, \(f\) models the predicted methane emissions in millions of tons produced by developed countries during year \(x .\) The function \(g\) models the same emissions for developing countries. $$\begin{array}{c|c|c|c|c|c}\boldsymbol{x} & 1990 & 2000 & 2010 & 2020 & 2030 \\\\\hline \boldsymbol{f}(\boldsymbol{x}) & 27 & 28 & 29 & 30 & 31 \\\\\hline \boldsymbol{g}(\boldsymbol{x}) & 5 & 7.5 & 10 & 12.5 & 15\end{array}$$ (a) Make a table for a function \(h\) that models the total predicted methane emissions for developed and developing countries. (b) Write an equation that relates \(f(x), g(x),\) and \(h(x)\)

Step by step solution

01

Understanding the Problem

We have two functions, \(f(x)\) and \(g(x)\), representing methane emissions in developed and developing countries respectively, for given years \(x\). The task is to find a new function \(h(x)\) that gives the total emissions by adding \(f(x)\) and \(g(x)\).

02

Create the Table for Function \(h\)

To create the table for \(h(x)\), add the values of \(f(x)\) and \(g(x)\) for each year \(x\) provided:- For 1990: \(h(1990) = f(1990) + g(1990) = 27 + 5 = 32\).- For 2000: \(h(2000) = f(2000) + g(2000) = 28 + 7.5 = 35.5\).- For 2010: \(h(2010) = f(2010) + g(2010) = 29 + 10 = 39\).- For 2020: \(h(2020) = f(2020) + g(2020) = 30 + 12.5 = 42.5\).- For 2030: \(h(2030) = f(2030) + g(2030) = 31 + 15 = 46\). The table for \(h(x)\) becomes:\[\begin{array}{c|c|c|c|c|c}\boldsymbol{x} & 1990 & 2000 & 2010 & 2020 & 2030 \hline\boldsymbol{h}(oldsymbol{x}) & 32 & 35.5 & 39 & 42.5 & 46 \end{array}\]

03

Writing the Equation

To relate \(f(x), g(x),\) and \(h(x)\), use the sum of the emissions as the relationship:\[h(x) = f(x) + g(x)\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Methane Emissions

Methane is a significant greenhouse gas that impacts our planet's climate. It allows sunlight to penetrate the atmosphere easily, but once inside, it efficiently traps heat, preventing it from escaping back into space. This "greenhouse effect" contributes to global warming. Methane is primarily released through activities like burning fossil fuels. As such, understanding and predicting methane emissions are crucial in addressing environmental concerns. In our problem, functions model the methane emissions for developed and developing countries. This helps track and predict the future impact by understanding where emissions have come from in past years.

Function Table

A function table is a handy tool used to display data points and visualize relationships between variables. In this case, the tables provided for functions \(f(x)\) and \(g(x)\) display methane emissions over various years for developed and developing countries, respectively. Each entry in the table corresponds to a year \(x\), and it gives the value of the function at that year. To find a new function \(h(x)\), which represents total emissions, you add the values of \(f(x)\) and \(g(x)\) for each year. This results in a new table showing the combined emissions from both types of countries.

Equation Writing

Writing equations is a fundamental skill in mathematics that involves defining relationships between different quantities. In this exercise, we define an equation to represent the total methane emissions. We were given two functions, \(f(x)\) and \(g(x)\), and our task was to find \(h(x)\), which sums these emissions. The method is straightforward: for each respective year \(x\), \(h(x) = f(x) + g(x)\). By using this equation, we can determine the total methane emissions across both developed and developing countries for any given year, illustrating the power of algebraic equations to model real-world data.