The table gives methane gas emissions in millions of metric tons. The quadratic model $y=0.0429 x^{2}-9.73 x+703$ approximates the emissions for these years. In the model, $x$ represents the number of years since 2008 , so $x=0$ represents $2008, x=1$ represents 2009 , and so on. Complete parts a and $b$.
\begin{tabular}{c|c}
Year & $\begin{array}{c}\text { Millions of Metric } \\
\text { Tons of Methane }\end{array}$ \\
\hline 2008 & 703.0 \\
\hline 2009 & 693.3 \\
\hline 2010 & 683.7 \\
\hline 2011 & 674.2 \\
\hline 2012 & 664.8 \\
\hline
\end{tabular}
a. According to the model, what would emissions be in 2013?
According to the model, in 2013 the emissions would be 655.4 million metric tons.
(Round to the nearest tenth as needed.)
b. Find the nearest year beyond 2008 for which this model predicts that emissions will reach $\mathbf{5 0 0}$ million metric tons.
The nearest year beyond 2008 for which the model predicts that the emissions will reach 500 million metric tons is
(Round down to the nearest year.)
Answer
Adding the result to 2008, we get the year as \(2008 + 23 = 2031\). So, the nearest year beyond 2008 for which the model predicts that the emissions will reach 500 million metric tons is \(\boxed{2031}\).
Step 1 ：Given the quadratic model \(y=0.0429 x^{2}-9.73 x+703\), where \(x\) represents the number of years since 2008.
Step 2 ：For part a, we need to find the emissions in 2013. Since 2013 is 5 years after 2008, we substitute \(x=5\) into the model.
Step 3 ：Calculating the model with \(x=5\), we get \(y = 0.0429*5^{2}-9.73*5+703 = 655.4\). So, according to the model, the emissions in 2013 would be approximately \(\boxed{655.4}\) million metric tons.
Step 4 ：For part b, we need to find the nearest year beyond 2008 when the emissions will reach 500 million metric tons. To do this, we set the model equal to 500 and solve for \(x\).
Step 5 ：Solving the equation \(0.0429 x^{2}-9.73 x+703 = 500\) for \(x\), we get two solutions \(x = 23.2458154659177\) and \(x = 203.560711340609\). We choose the smaller value because we are looking for the nearest year beyond 2008.
Step 6 ：Adding the result to 2008, we get the year as \(2008 + 23 = 2031\). So, the nearest year beyond 2008 for which the model predicts that the emissions will reach 500 million metric tons is \(\boxed{2031}\).
