360 CHAPTER 5 Exponential and Logarithmic Functions (0, P(0)) c1 – 2 t P(t ) y 5 c Inflection point Logistic decay (b) P(t) 5 , b , 0 c 1 1 ae2bt Figure 58 (0, P(0)) t P(t ) y 5 c c1 – 2 Inflection point (a) P(t) 5 , b . 0 c 1 1 ae2bt Logistic growth (b) Figure 57(a) shows the graph of ( ) = + − u t e 30 70 t 0.0673 for ≥ t 0 using Geogebra. (c) Graph ( ) = + − u x e 30 70 x 0.0673 and = y 50 where x is time using Geogebra. Find the intersection point. See Figure 57(b). It takes = t 18.6 minutes (18 minutes, 37 seconds) for the temperature to cool to ° 50 C. Figure 57 ( ) = + − u t e 30 70 t 0.0673 (b) (a) (b) (d) Graph ( ) = + − u x e 30 70 x 0.0673 and = y 35, where x is time. Use Geogebra to find that it takes = x 39.21 minutes (39 minutes, 13 seconds) for the temperature to cool to ° 35 C. (e) As t increases, the value of −e t 0.0673 approaches zero, so the value of u, the temperature of the object, approaches ° 30 C, the air temperature of the room. Now Work PROBLEM 13 4 Use Logistic Models The exponential growth model ( ) = > A t A e k , 0, kt 0 assumes uninhibited growth, meaning that the value of the function grows without limit. Recall that cell division could be modeled using this function, assuming that no cells die and no by-products are produced. However, cell division eventually is limited by factors such as living space and food supply. The logistic model , given next, can describe situations where the growth or decay of the dependent variable is limited. Logistic Model In a logistic model, the population P after time t is given by the function ( ) = + − P t c ae 1 bt (6) where a, b, and c are constants with > a 0 and > c 0. The model is a growth model if > b 0; the model is a decay model if < b 0. The number c is called the carrying capacity (for growth models) because the value ( ) P t approaches c as t approaches infinity; that is, ( ) = →∞ P t c lim . t The number b is the growth rate for > b 0 and the decay rate for < b 0. Figure 58(a) shows the graph of a typical logistic growth function, and Figure 58(b) shows the graph of a typical logistic decay function.
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