Logistic Curve Problem, Algebraically: For unrestrained population growth, the rate of

Chapter 9, Problem 24

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Logistic Curve Problem, Algebraically: For unrestrained population growth, the rate of change of population is directly proportional to the population. That is, dp/dt = kp, where p is population, t is time, and k is a constant. One assumption for restrained growth is that there is a certain maximum size, m, for the population, and the rate goes to zero as the population approaches that size. Use this information to answer these questions. a. Show that the differential equation dp/dt = kp(m p) has the properties mentioned. b. At what value of p is the growth rate the greatest? c. Separate the variables, then solve the equation by integrating. If you have worked correctly, you can evaluate the integral on one side of the equation by partial fractions. d. Transform your answer so that p is explicitly in terms of t. Show that you can express it in the form of the logistic equation,OBJECTIVE Integrate (antidifferentiate) each of the six inverse trigonometric functions.where p0 is the population at time t = 0 andb is a constant e. Census figures for the United States are1960: 179.3 million1970: 203.2 million1980: 226.5 million1990: 248.7 millionLet t be time, in years, that has elapsed since1960. Use these data as initial conditions toevaluate p0, b, and k. Write the particularsolution.f. Predict the outcome of the 2000 census.How close does your answer come to theactual 2000 population, 281.4 million?g. Based on the logistic model, what will be theultimate U.S. population? h. Is this mathematical model very sensitive tothe initial conditions? For instance, supposethat the 1970 population had really been204.2 million instead of 203.2 million. Howmuch would this affect the predictedultimate population?Stanton and Irene Katchatag of Unalakeet,Alaska, were the first two people counted for the2000 U.S. census.