Problem
The capacities at which U.S. nuclear power plants are working are shown in table for various years. \begin{tabular}{|c|c|} \hline Year & Percent \\ \hline 1975 & 56 \\ \hline 1980 & 59 \\ \hline 1985 & 58 \\ \hline 1990 & 70 \\ \hline 1995 & 76 \\ \hline 2000 & 88 \\ \hline 2004 & 89 \\ \hline \end{tabular} Let $f(t)$ be the capacity (in percent) at which U.S. nuclear power plants are working at t years since 1970 . A model of the situation is $f(t)=0.027 t^{2}+0.216 t+53.296$. Use a graphing calculator to draw the graph of the model and, in the same viewing window, the scattergram of the data. Does the model fit the data well? The function is not a good model for the data The function is a good model for the data. Estimate at what capacity U. S. nuclear power plants were working in 2012. $\%$ Round to the nearest whole percent. Predict when U. S. nuclear power plants will be working at full (100\%) capacity. Enter the year this occurs.
Solution
Step 1 ：\(f(t) = 0.027t^2 + 0.216t + 53.296\)
Step 2 ：For 2012, \(t = 2012 - 1970 = 42\)
Step 3 ：Calculate \(f(42) = 0.027(42)^2 + 0.216(42) + 53.296\)
Step 4 ：\(f(42) = 0.027(1764) + 9.072 + 53.296\)
Step 5 ：\(f(42) = 47.628 + 9.072 + 53.296\)
Step 6 ：\(f(42) = 110.996\)
Step 7 ：\(\boxed{111\%}\)
Step 8 ：For 100\% capacity, \(f(t) = 100\)
Step 9 ：Solve \(0.027t^2 + 0.216t + 53.296 = 100\)
Step 10 ：\(0.027t^2 + 0.216t - 46.704 = 0\)
Step 11 ：Using a graphing calculator, find the approximate value of t
Step 12 ：\(t \approx 45.5\)
Step 13 ：Year = 1970 + 45.5
Step 14 ：\(\boxed{2015}\)
