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Radioactive decay: A scientist is studying the amount of a radioactive substance present over a period of time. A plot of the logarithm of the amount shows a linear pattern. What type of function should the scientist use to model the original data?

Unit conversion with exponential decay: The exponential function \(N=500 \times 0.68^{t}\), where \(t\) is measured in years, shows the amount, in grams, of a certain radioactive substance present. a. Calculate \(N(2)\) and explain what your answer means. b. What is the yearly percentage decay rate? c. What is the monthly decay factor rounded to three decimal places? What is the monthly percentage decay rate? d. What is the percentage decay rate per second? (Note: For this calculation, you will need to use all the decimal places that your calculator can show.)

Inflation: The yearly inflation rate tells the percentage by which prices increase. For example, from 1990 through 2000 the inflation rate in the United States remained stable at about \(3 \%\) each year. In 1990 an individual retired on a fixed income of \(\$ 36,000\) per year. Assuming that the inflation rate remains at \(3 \%\), determine how long it will take for the retirement income to deflate to half its 1990 value. (Note: To say that retirement income has deflated to half its 1990 value means that prices have doubled.)

Cell phones: The following table shows the number, in millions, of cell phone subscribers in the United States at the end of the given year. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Subscribers (millions) } \\ \hline 2001 & 128.4 \\ \hline 2002 & 140.8 \\ \hline 2003 & 158.7 \\ \hline 2004 & 182.1 \\ \hline 2005 & 207.9 \\ \hline \end{array} $$ a. Plot the natural logarithm of the data points. Does this plot make it look reasonable to approximate the original data with an exponential function? b. Find the regression line for the natural logarithm of the data and add its graph to the plot in part a. c. Construct an exponential model for the original subscribership data using the logarithm as a link.

Nearly linear or exponential data: One of the two tables below shows data that are better approximated with a linear function, and the other shows data that are better approximated with an exponential function. Make plots to identify which is which, and then use the appropriate regression to find models for both. $$ \begin{aligned} &\begin{array}{|c|c|} \hline t & f(t) \\ \hline 1 & 3.62 \\ \hline 2 & 23.01 \\ \hline 3 & 44.26 \\ \hline 4 & 62.17 \\ \hline 5 & 83.25 \\ \hline \end{array}\\\ &\begin{array}{|c|c|} \hline t & g(t) \\ \hline 1 & 3.62 \\ \hline 2 & 5.63 \\ \hline 3 & 8.83 \\ \hline 4 & 13.62 \\ \hline 5 & 21.22 \\ \hline \end{array} \end{aligned} $$

. The half-life of U239: Uranium 239 is an unstable isotope of uranium that decays rapidly. In order to determine the rate of decay, 1 gram of U239 was placed in a container, and the amount remaining was measured at 1-minute intervals and recorded in the table below $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Time } \\ \text { in minutes } \end{array} & \begin{array}{c} \text { Grams } \\ \text { remaining } \end{array} \\ \hline 0 & 1 \\ \hline 1 & 0.971 \\ \hline 2 & 0.943 \\ \hline 3 & 0.916 \\ \hline 4 & 0.889 \\ \hline 5 & 0.863 \\ \hline \end{array} $$ a. Show that these are exponential data and find an exponential model. (For this problem, round all your answers to three decimal places.) b. What is the percentage decay rate each minute? What does this number mean in practical terms? c. Use functional notation to express the amount remaining after 10 minutes and then calculate that value. d. What is the half-life of U239?

Atmospheric pressure: The table below gives a measurement of atmospheric pressure, in grams per square centimeter, at the given altitude, in kilometers.17 $$ \begin{array}{|c|c|} \hline \text { Altitude } & \text { Atmospheric pressure } \\ \hline 5 & 569 \\ \hline 10 & 313 \\ \hline 15 & 172 \\ \hline 20 & 95 \\ \hline 25 & 52 \\ \hline \end{array} $$ (For comparison, 1 kilometer is about 0.6 mile, and 1 gram per square centimeter is about 2 pounds per square foot.) a. Plot the data on atmospheric pressure. b. Make an exponential model for the data on atmospheric pressure. c. What is the atmospheric pressure at an altitude of 30 kilometers? d. Find the atmospheric pressure on Earth’s surface. This is termed standard atmospheric pressure. e. At what altitude is the atmospheric pressure equal to 25% of standard atmospheric pressure?

Long-term population growth: Although exponential growth can often be used to model population growth accurately for some periods of time, there are inevitably, in the long term, limiting factors that make purely exponential models inaccurate. If the U.S. population had continued to grow by \(3 \%\) each year from 1790 , when it was \(3.93\) million, until today, what would the population of the United States have been in 2000 ? For comparison, according to census data, the population of the United States in 2000 was \(281,421,906\). The population of the world was just over 6 billion people.

Wages: A worker is reviewing his pay increases over the past several years. The table below shows the hourly wage W, in dollars, that he earned as a function of time t, measured in years since the beginning of 1990. $$ \begin{array}{|c|c|} \hline \text { Time } t & \text { Wage } W \\ \hline 1 & 15.30 \\ \hline 2 & 15.60 \\ \hline 3 & 15.90 \\ \hline 4 & 16.25 \\ \hline \end{array} $$ a. By calculating ratios, show that the data in this table are exponential. (Round the quotients to two decimal places.) b. What is the yearly growth factor for the data? c. The worker can't remember what hourly wage he earned at the beginning of 1990 . Assuming that \(W\) is indeed an exponential function, determine what that hourly wage was. d. Find a formula giving an exponential model for \(W\) as a function of \(t\). e. What percentage raise did the worker receive each year?f. Given that prices increased by 34% over the decade of the 1990s, use your model to determine whether the worker’s wage increases kept pace with inflation.

$$ \begin{array}{|c|c|c|c|} \hline \text { Year } & \text { Wolves } & \text { Year } & \text { Wolves } \\\ \hline 1985 & 15 & 1993 & 40 \\ \hline 1986 & 16 & 1994 & 57 \\ \hline 1987 & 18 & 1995 & 83 \\ \hline 1988 & 28 & 1996 & 99 \\ \hline 1989 & 31 & 1997 & 145 \\ \hline 1990 & 34 & 1998 & 178 \\ \hline 1991 & 40 & 1999 & 197 \\ \hline 1992 & 45 & 2000 & 266 \\ \hline \end{array} $$ Gray wolves in Wisconsin: Gray wolves were among the first mammals protected under the Endangered Species Act in the \(1970 \mathrm{~s}\). Wolves recolonized in Wisconsin beginning in 1980 . Their population has grown reliably since 1985 as follows: \({ }^{21}\) a. Explain why an exponential model may be appropriate. b. Are these data exactly exponential? Explain. c. Find an exponential model for these data. d. Plot the data and the exponential model. e. Comment on your graph in part d. Which data points are below or above the number predicted by the exponential model?