SECTION 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 361 Based on the figures, the following properties of logistic functions emerge. Properties of the Logistic Model, Equation (6) • The domain is the set of all real numbers. The range is the interval ( )c 0, where c is the carrying capacity. • There are no x -intercepts; the y -intercept is ( ) P 0 . • There are two horizontal asymptotes: = y 0 and = y c. • ( ) P t is an increasing function if > b 0 and a decreasing function if < b 0. • There is an inflection point where ( ) P t equals 1 2 of the carrying capacity. The inflection point is the point on the graph where the graph changes from being concave up to being concave down for growth functions, and the point where the graph changes from being concave down to being concave up for decay functions. At the inflection point, the rate of change of P is a maximum and the graph is steepest. In other words, the function P is growing (or decaying) the fastest at the inflection point. • The graph is smooth and continuous, with no corners or gaps. Fruit Fly Population Fruit flies are placed in a half-pint milk bottle with a banana (for food) and yeast plants (for food and to provide a stimulus to lay eggs). Suppose that the fruit fly population after t days is given by ( ) = + − P t e 230 1 56.5 t 0.37 (a) State the carrying capacity and the growth rate. (b) Determine the initial population. (c) What is the population after 5 days? (d) How long does it take for the population to reach 180? (e) Use a graphing utility to determine how long it takes for the population to reach one-half of the carrying capacity. Solution EXAMPLE 5 (a) As →∞ → − t e , 0 t 0.37 and ( ) → P t 230 1 . The carrying capacity of the half-pint bottle is 230 fruit flies. The growth rate is = = b 0.37 37% per day. (b) To find the initial number of fruit flies in the half-pint bottle, evaluate ( ) P 0 . ( ) = + = + = − ⋅ P e 0 230 1 56.5 230 1 56.5 4 0.37 0 So, initially, there were 4 fruit flies in the half-pint bottle. (c) After 5 days the number of fruit flies in the half-pint bottle is ( ) = + ≈ − ⋅ P e 5 230 1 56.5 23 fruit flies 0.37 5 After 5 days, there are approximately 23 fruit flies in the bottle. (d) To determine when the population of fruit flies will be 180, solve the equation ( ) ( ) = + = = + ≈ ≈ − − − − P t e e e e 180 230 1 56.5 180 230 180 1 56.5 0.2778 56.5 0.2778 56.5 t t t t 0.37 0.37 0.37 0.37 Divide both sides by 180. Subtract 1 from both sides. (continued)
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