Problem
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 Check answer
Solution
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.
