PK P (K-P)e 8. Scientists use the Logistic Growth function P(t) = to model population growth where P theoretical upper bound of the population and population is the population at some reference point, K is the carrying capacity which is the base growth rate of the 1s a Write a logistic growth function for the world population in billions if the population in 1999 (t 0) reached 6 billion, K = 15 billion and , = 0.025 per year. a. b. Use technology to graph P(t) on the interval [0, 200] x [0, 15]. Zoom in or out to see they apply to this problem. over the given interval. Be sure to label axes as What will the population be in the year 2020? c. d. When will the population reach 12 billion? Find the growth rate function of the world population. Be sure to show all steps. e.

PK P (K-P)e 8. Scientists use the Logistic Growth function P(t) = to model population growth where P theoretical upper bound of the population and population is the population at some reference point, K is the carrying capacity which is the base growth rate of the 1s a Write a logistic growth function for the world population in billions if the population in 1999 (t 0) reached 6 billion, K = 15 billion and , = 0.025 per year. a. b. Use technology to graph P(t) on the interval [0, 200] x [0, 15]. Zoom in or out to see they apply to this problem. over the given interval. Be sure to label axes as What will the population be in the year 2020? c. d. When will the population reach 12 billion? Find the growth rate function of the world population. Be sure to show all steps. e.

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter4: Exponential And Logarithmic Functions

Section4.2: The Natural Exponential Function

Problem 27E: Logistic Growth Animal populations are not capable of unrestricted growth because of limited habitat...

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Eastern Pacific Yellowfin Tuna Studies to fit a logistic model to the Eastern Pacific yellowfin tuna population have yielded N=1481+36e2.61t where t is measured in years and N is measured in thousands of tons of fish. a. What is the r value for the Eastern Pacific yellowfin tuna? b. What is the carrying capacity K for the Eastern Pacific yellowfin tuna? c. What is the optimum yield level? d. Use your calculator to graph N versus t. e. At what time was the population growing the most rapidly?
More on the Pacific Sardine This is a continuation of Example 5.1. In this exercise, we explore the Pacific sardine population further, using the model in Example 5.1. a. If the current level of the Pacific sardine population is 50,000 tons, how long will it take for the population to recover to the optimum growth level of 1.2milliontons? Suggestion: One way to solve this is to make a new logistic formula using K2.4, r0.338, and N(0)0.05. b. The value of r used in Example 5.1 ignores the effects of fishing. If fishing mortality is taken into account, then r drops to 0.215 per year with the carrying capacity still at 2.4milliontons. Answer the question in part a using this lower value of r. Note: The population estimate of 50,000 tons and the adjusted value of r are given in the paper by Murphy see footnote 3 on page 347. Murphy points out that factoring in the growth of the competing anchovy population makes the recovery times even longer, and he adds. "It is disconcerting to realize how slowly the population will recover to its level of maximum productivity ... even if fishing stops." Studies to fit a logistic model to the Pacific sardine population have yielded. N=241+239e0.338t where t is measured in years and N is measured in millions of tons of fish. Part 1 What is r for the Pacific sardine? Part 2 According to the logistic model, in the absence of limiting factors, what would be the annual percentage growth rate for the Pacific sardine? Part 3 What is the environmental carrying capacity K? Part 4 What is the optimum yield level? Part 5 Make a graph of N versus t. Part 6 At what time t should the population he harvested? Part 7 What portion of the graph is concave up? What portion is concave down?
Modeling Human Height with a Logistic Function A male child is 21inches long at birth and grows to an adult height of 73inches. In this exercise, we make a logistic model of his height as a function of age. a. Use the given information to find K and b for the logistic model. b. Suppose he reaches 95 of his adult height at age 16. Use this information and that from part a to find r. Suggestion: You will need to use either the crossing-graphs method or some algebra involving the logarithm. c. Make a logistic model for his height H, in inches, as a function of his age t, in years. d. According to the logistic model, at what age is he growing the fastest? e. Is your answer to part d consistent with your knowledge of how humans grow?
Population The table shows the mid-year populations (in millions) of five countries in 2015 and the projected populations (in millions) for the year 2025. (a) Find the exponential growth or decay model y=aebt or y=aebt for the population of each country by letting t=15 correspond to 2015. Use the model to predict the population of each country in 2035. (b) You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation y=aebt gives the growth rate? Discuss the relationship between the different growth rates and the magnitude of the constant.
The table shows the mid-year populations (in millions) of five countries in 2015 and the projected populations (in millions) for the year 2025. (a) Find the exponential growth or decay model y=aebt or y=aebt for the population of each country by letting t=15 correspond to 2015. Use the model to predict the population of each country in 2035. (b) You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation y=aebt gives the growth rate? Discuss the relationship between the different growth rates and the magnitude of the constant.
Bird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model n(t)=56000.5+27.5e0.044t where t is measured in years. a Find the initial bird population. b Draw a graph of the function n(t). c What size does the population approach as time goes on?
Logistic Growth Animal populations are not capable of unrestricted growth becomes of limited habitat and food supplies. Under such conditions the population follows a logistic growth model: P(t)=d1+kect where c, d, and k are positive constants. For a certain fish population is a small pond d=1200,k=11,c=0.2, and t is measured in years. The fish were introduced into the pond at time t=0. a How many fish were originally put in the pond? b Find the population after 10, 20, and 30 years. c Evaluate P(t) for large values of t. What value does the population approach as t? Does the graph show confirm your calculations?
Bird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model p(t)=73.26.1+5.9e0.02r Where t is measured in years. Find the initial bird population. Draw a graph of the function n(t) . What size does the population approach as time goes on?
Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and bacteria initiates the decay process. Let A be the amount of litter present, in grams per square meter, as a function of time t in years. If the litter falls at a constant rate of L grams per square meter per year, and if it decays at a constant proportional rate of k per year, then the limiting value of A is R=L/k. For this exercise and the next, we suppose that at time t=0, the forest floor is clear of litter. a. If D is the difference between the limiting value and A, so that D=RA, then D is an exponential function of time. Find the initial value of D in terms of R. b. The yearly decay factor for D is ek. Find a formula for D in term of R and k. Reminder:(ab)c=abc. c. Explain why A=RRekt.