The percent of paper and paperboard recycled from municipal solid waste in a certain town has grown exponentially, as shown in the table to the right. Use a graphing calculator to fit the data to an exponential function. Use the function to find $R(x)$ in 2007.
Type the values of $a$ and $b$ from your calculator for the exponential function $y=a b^{x}$.
\[
\begin{array}{l}
a=\square \\
b=\square
\end{array}
\]
(Round to six decimal places as needed.)
\begin{tabular}{|c|c|}
\hline & \begin{tabular}{c}
Percent of \\
paper recycled, \\
$\mathrm{R}(\mathbf{x})$
\end{tabular} \\
\hline Year, $\mathbf{x}$ & \begin{tabular}{c}
(1980,0 \\
1990,10
\end{tabular} \\
1994,14 & 27.6 \\
1995,15 & 35.4 \\
1996,16 & 40.1 \\
1997,17 & 40.5 \\
1998,18 & 40.4 \\
1999,19 & 40.9 \\
2000,20 & 41.2 \\
\hline
\end{tabular}
Clear all
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Answer
Remember to round your final answers to six decimal places as needed.
Step 1 ：Input the given data into your graphing calculator. The x-values are the years (subtract 1980 from each year to make calculations easier), and the y-values are the percent of paper recycled.
Step 2 ：Use the calculator's regression function to fit the data to an exponential function. This function is often labeled as 'ExpReg' or something similar on most graphing calculators.
Step 3 ：The calculator will output the values of \(a\) and \(b\) for the function \(y=a b^{x}\). These are the values you're looking for.
Step 4 ：To find \(R(x)\) in 2007, simply plug in \(x=2007-1980=27\) into the function \(y=a b^{x}\) and calculate the result.
Step 5 ：Remember to round your final answers to six decimal places as needed.
