372 CHAPTER 5 Exponential and Logarithmic Functions (a) Using a graphing utility, draw a scatter plot of the data using years since 1900 as the independent variable and population as the dependent variable. (b) Using a graphing utility, build a logistic model from the data. (c) Using a graphing utility, graph the function found in part (b) on the scatter plot. (d) Based on the function found in part (b), what is the carrying capacity of the United States? (e) Use the function found in part (b) to predict the population of the United States in 2025. (f) When will the United States population be 375 million? 8. Population Model The data that follow represent world population. An ecologist is interested in building a model that describes the world population. Year Population (billions) 6.96 7.04 7.13 7.21 7.30 7.38 7.47 7.55 2010 2011 2012 2013 2014 2015 2016 2017 Year Population (billions) 2018 2019 2020 2021 2022 2023 2024 7.63 7.71 7.84 7.91 7.98 8.05 8.12 Source: worldometers.info (x = 0) (x = 1) (x = 2) (x = 3) (x = 4) (x = 5) (x = 6) (x = 7) (x = 8) (x = 9) (x = 10) (x = 11) (x = 12) (x = 13) (x = 14) (a) Using a graphing utility, draw a scatter plot of the data using years since 2010 as the independent variable and population as the dependent variable. (b) Using a graphing utility, build a logistic model from the data. (c) Using a graphing utility, graph the function found in part (b) on the scatter plot. (d) Based on the function found in part (b), what is the carrying capacity of the world? (e) Use the function found in part (b) to predict the population of the world in 2025. (f) When will world population be 9 billion? 9. Mixed Practice Chirping Crickets Crickets make a chirping noise by sliding their wings rapidly over each other. Perhaps you have noticed that the number of chirps seems to increase with the temperature. The data below represent the temperature (in degrees Fahrenheit) and the number of chirps per second for the striped ground cricket. Temperature, x Chirps per Second, y Temperature, x Chirps per Second, y 88.6 20.0 71.6 16.0 93.3 19.8 84.3 18.4 80.6 17.1 75.2 15.5 69.7 14.7 82.0 17.1 69.4 15.4 83.3 16.2 79.6 15.0 82.6 17.2 80.6 16.0 83.5 17.0 76.3 14.4 Source: George W. Pierce. The Songs of Insects. Cambridge, MA: Harvard University Press, 1949, pp. 12–21. 6. Social Networking The data in the table below represent the percent of U.S. citizens aged 12 and older who have a profile on at least one social network. Year Percent on a Social Networking Site 2008 ( ) = x 1 10 2009 ( ) = x 2 21 2010 ( ) = x 3 44 2011 ( ) = x 4 53 2012 ( ) = x 5 57 2013 ( ) = x 6 63 2014 ( ) = x 7 66 2015 ( ) = x 8 70 2016 ( ) = x 9 77 2017 ( ) = x 10 80 2018 ( ) = x 11 77 2019 ( ) = x 12 79 2020 ( ) = x 13 80 2021 ( ) = x 14 82 2022 ( ) = x 15 91 2023 ( ) = x 16 93 Source: Statista.com, 2024 (a) Using a graphing utility, draw a scatter plot of the data using 8 for 2008, 9 for 2009, and so on, as the independent variable, and percent on social networking site as the dependent variable. (b) Using a graphing utility, build a logarithmic model from the data. (c) Graph the logarithmic function found in part (b) on the scatter plot. (d) Use the model to predict the percent of U.S. citizens on social networking sites in 2020. (e) Use the model to predict the year in which 99% of U.S. citizens will be on social networking sites. 7. Population Model The following data represent the population of the United States.An ecologist is interested in building a model that describes the population of the United States. Source: U.S. Census Bureau Population (Millions) Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 (x = 0) (x = 10) (x = 20) (x = 30) (x = 40) (x = 50) (x = 60) (x = 70) (x = 80) (x = 90) (x = 100) (x = 110) (x = 120) 76.2 92.2 106.0 123.2 132.2 151.3 179.3 203.3 226.5 248.7 281.4 308.7 331.4
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