SECTION 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 365 and set free. Based on experience, the environmentalists expect the population to grow according to the model ( ) = + − P t e 500 1 83.33 t 0.162 where t is measured in years. Credit: Jupiterimages/Getty Images (a) Determine the carrying capacity of the environment. (b) What is the growth rate of the bald eagle? (c) Use a graphing utility to graph ( ) = P P t . (d) What is the population after 3 years? (e) When will the population be 300 eagles? (f) How long does it take for the population to reach onehalf of the carrying capacity? 27. Invasive Species A habitat can be altered by invasive species that crowd out or replace native species. The logistic model ( ) = + − P t e 431 1 7.91 t 0.017 represents the number of invasive species present in the Great Lakes t years after 1900. (a) Evaluate and interpret ( ) P 0 . (b) What is the growth rate of invasive species? (c) Use a graphing utility to graph ( ) = P P t . (d) How many invasive species were present in the Great Lakes in 2000? (e) In what year was the number of invasive species 175? Source: NOAA 28. Social Networking The logistic model ( ) = + − P t e 86.1 1 2.12 t 0.361 gives the percentage of people in the United States who have a social media profile, where t represents the number of years after 2008. (a) Evaluate and interpret ( ) P 0 . (b) What is the growth rate? (c) Use a graphing utility to graph ( ) = P P t . (d) During 2017, what percentage of people in the United States had a social media profile? (e) In what year did 69.3% of people in the United States have a social media profile? Source: Statista, 2023 23. Tablet Computers The logistic model ( ) = + − P t e 50.9249 1 14.9863 t 1.0404 represents the percentage of U.S. households that own a tablet computer t years after 2010. (a) Evaluate and interpret P(0). (b) Use a graphing utility to graph ( ) = P P t . (c) What percentage of U.S. households owned a tablet computer in 2018. (d) In what year did the percentage of U.S. households that owned a tablet computer reach 47%? 24. Farmers The logistic model ( ) = + W t e 14,656,248 1 0.059 t 0.057 represents the number of farm workers in the United States t years after 1910. (a) Evaluate and interpret ( ) W 0 . (b) Use a graphing utility to graph ( ) = W W t . (c) How many farm workers were there in the United States in 2010? (d) When did the number of farm workers in the United States reach 10,000,000? (e) According to this model, what happens to the number of farm workers in the United States as t approaches ∞? Based on this result, do you think that it is reasonable to use this model to predict the number of farm workers in the United States in 2060? Why? Source: U.S. Department of Agriculture 25. Birthdays The logistic model ( ) = + P n e 113.3198 1 0.115 n 0.0912 models the probability that, in a room of n people, no two people share the same birthday. (a) Use a graphing utility to graph ( ) = P P n . (b) In a room of = n 15 people, what is the probability that no two share the same birthday? (c) How many people must be in a room before the probability that no two people share the same birthday falls below 10%? (d) What happens to the probability as n increases? Explain what this result means. 26. Population of an Endangered Species Environmentalists often capture an endangered species and transport the species to a controlled environment where the species can produce offspring and regenerate its population. Suppose that six American bald eagles are captured, transported to Montana, Problems 29 and 30 use the following discussion: Uninhibited growth can be modeled by exponential functions other than ( ) = A t A e . kt 0 For example, if an initial population P0 requires n units of time to double, then the function ( ) = ⋅ P t P 2t n 0 models the size of the population at time t. Likewise, a population requiring n units of time to triple can be modeled by ( ) = ⋅ P t P 3 . t n 0 29. Growth of an Insect Population An insect population grows exponentially. (a) If the population triples in 20 days, and 50 insects are present initially, write an exponential function of the form ( ) = ⋅ P t P 3t n 0 that models the population. (b) Graph the function using a graphing utility. (c) What will the population be in 47 days? (d) When will the population reach 700? (e) Express the model from part (a) in the form ( ) = A t A e . kt 0 30. Growth of a Human Population The population of a town is growing exponentially. (a) If its population doubled in size over an 8-year period and the current population is 25,000, write an exponential function of the form ( ) = ⋅ P t P 2t n 0 that models the population. (b) Graph the function using a graphing utility. (c) What will the population be in 3 years? (d) When will the population reach 80,000? (e) Express the model from part (a) in the form ( ) = A t A e . kt 0
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