The table gives methane gas emissions from all sources in...

The table gives methane gas emissions from all sources in the United States, in millions of metric tons. The quadratic model $$y=1.493 x^{2}+7.279 x+684.4$$ approximates the emissions for these years. In the model, $x$ represents the number of years since 2004 so $x=0$ represents $2004, x=1$ represents 2005 and so on. (a) According to the model, what would be the emissions in $2010 ?$ Round to the nearest tenth of a million metric tons. (b) Find the year beyond 2004 for which this model predicts that the emissions reached 700 million metric tons. $$\begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Millions of Metric } \\ \text { Tons of Methane } \end{array} \\ \hline 2004 & 686.6 \\ \hline 2005 & 691.8 \\ \hline 2006 & 706.3 \\ \hline 2007 & 722.7 \\ \hline 2008 & 737.4 \\ \hline \end{array}$$

The table gives methane gas emissions from all sources in the United States, in millions of metric tons. The quadratic model
$$y=1.493 x^{2}+7.279 x+684.4$$
approximates the emissions for these years. In the model, $x$ represents the number of years since 2004 so $x=0$ represents $2004, x=1$ represents 2005 and so on.
(a) According to the model, what would be the emissions in $2010 ?$ Round to the nearest tenth of a million metric tons.
(b) Find the year beyond 2004 for which this model predicts that the emissions reached 700 million metric tons.
$$\begin{array}{|c|c|}
\hline \text { Year } & \begin{array}{c}
\text { Millions of Metric } \\
\text { Tons of Methane }
\end{array} \\
\hline 2004 & 686.6 \\
\hline 2005 & 691.8 \\
\hline 2006 & 706.3 \\
\hline 2007 & 722.7 \\
\hline 2008 & 737.4 \\
\hline
\end{array}$$

Key Concepts

Interpreting Variables in Models

Interpreting variables in mathematical models involves understanding what each variable represents in the real-world context. This allows one to translate the mathematical solution back into a meaningful answer, such as determining which time period or scenario an equation is describing.

Solving Quadratic Equations

Solving quadratic equations entails finding the values of the variable that make the quadratic expression equal to zero or another given value. Techniques such as factoring, completing the square, or applying the quadratic formula are used. This concept is crucial for determining the points in time or conditions at which the model’s predictions meet certain criteria.

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable into a function to determine the corresponding output. This process is essential in modeling as it allows one to make predictions or compute expected outcomes at particular points in time or under specific conditions.

Quadratic Functions

A quadratic function is a second-degree polynomial function of the form ax² + bx + c, which graphically represents a parabola. In modeling, quadratic functions are used to capture relationships where the rate of change itself changes at a non-constant rate, often leading to curves that open upward or downward. This concept is key when the behavior of the modeled phenomenon accelerates over time.

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