The table gives the number of active Twitter users worldwide, semiannually from 2010 to 2016. \begin{array}{|c|c|} \hline \text{Years since} & \text{January 1, 2010} & \text{Twitter user} & \text{(millions)} \\ \hline 0 & 30 & 3.5 & 232 \\ \hline 0.5 & 49 & 4.0 & 255\\ \hline 1.0 & 68 & 4.5 & 284 \\ \hline 1.5 & 101 & 5.0 & 302 \\ \hline 2.0 & 138 & 5.5 & 307 \\ \hline 2.5 & 167 & 6.0 & 310 \\ \hline 3.0 & 204 & 6.5 & 317 \\ \hline \end{array}

Use a calculator or computer to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models.

The table gives the midyear population of Japan, in thousands, from 1960 to 2010.
\(\begin{array}{|c|c|}\hline \text{Year} & \text{Population} \\ \hline 1960 & 94.092 \\ \hline 1965 & 98.883 \\ \hline 1970 & 104.345 \\ \hline 1975 & 111.573 \\ \hline 1980 & 116.807 \\ \hline 1985 & 120.754 \\ \hline 1990 & 123.537 \\ \hline 1995 & 125.327 \\ \hline 2000 & 126.776 \\ \hline 2005 & 127.715 \\ \hline 2010 & 127.579 \\ \hline \end{array}\)
Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 94,000 from each of the population figures. Then, after obtaining a model from your calculator, add 94,000 to get your final model. It might be helpful to choose \(t=0\) to correspond to 1960 or 1980.]

The annual sales S (in millions of dollars) for the Perrigo Company from 2004 through 2010 are shown in the table. \(\begin{array}{|c|c|} \hline \text{Year} & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\ \hline \text{Sales, S} & 898.2 & 1024.1 & 1366.8 & 1447.4 & 1822.1 & 2006.9 & 2268.9\\ \hline \end{array}\)

a) Use a graphing utility to create a scatter plot of the data. Let t represent the year, with t=4 corresponding to 2004. b) Use the regression feature of the graphing utility to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln\ b}\)to rewrite the model as an exponential model in base e. c) Use the regression feature of the graphing utility to find a logarithmic model for the data. d) Use the exponential model in base e and the logarithmic model to predict sales in 2011. It is projected that sales in 2011 will be $2740 million. Do the predictions from the two models agree with this projection? Explain.