495 4.5 Exponential and Logarithmic Equations Applications and Models EXAMPLE 10 Applying an Exponential Equation to the Strength of a Habit The strength of a habit is a function of the number of times the habit is repeated. If N is the number of repetitions and H is the strength of the habit, then, according to psychologist C. L. Hull, H= 100011 - e-kN2, where k is a constant. Solve this equation for k. SOLUTION H= 100011 - e-kN2 H 1000 = 1 - e-kN Divide by 1000. H 1000 - 1 = -e-kN Subtract 1. e-kN = 1 - H 1000 Multiply by -1 and rewrite. ln e-kN = lna1 - H 1000b Take the natural logarithm on each side. -kN = lna1 - H 1000b lnex = x k = - 1 N lna1 - H 1000b Multiply by -1 N. With the final equation, if one pair of values for H and N is known, k can be found, and the equation can then be used to find either H or N for given values of the other variable. S Now Try Exercise 91. Now solve for k. First solve for e-kN. EXAMPLE 11 Modeling Viewers of Video Content The table gives the number of U.S. residents (in millions) who watched video content on a tablet each year from 2013 through 2020. The data can be modeled by the function ƒ1t2 = 23.19 ln t + 75.19, t Ú 1, where t is the number of years after 2012. (a) Use the function to estimate the number of U.S. residents, to the nearest tenth of a million, who watched video content on a tablet in 2019. (b) If this trend continues, approximately when will the number of U.S. residents who watch video content on a tablet reach 130 million? SOLUTION (a) The year 2019 is represented by t = 2019 - 2012 = 7. ƒ1t2 = 23.19 ln t + 75.19 Given function ƒ172 = 23.19 ln 7 + 75.19 Let t = 7. ƒ172 ≈120.3 Use a calculator. Based on this model, 120.3 million U.S. residents watched video content on a tablet in 2019. Year Tablet Video Viewers (in millions) 2013 76.7 2014 89.0 2015 100.8 2016 107.2 2017 112.6 2018 116.5 2019 120.5 2020 124.2 Data from www.statista.com
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