The Logistic Function (Population Growth)

Calculus Level 3

Exponential functions are prevalent in many problems in science, business, economics, medicine, and sociology. One important function, the logistic function, is often used to analyze population growth that is limited by natural environmental factors. The general form of the logistic function is

P(t)=A1+BeCt, P(t)=\frac{A}{1+Be^{-Ct}}, where constants AA, BB, and CC are usually determined experimentally.

Suppose that population in a town is modeled by

P(t)=20,0001+4e2t, P(t)=\frac{20,000}{1+4e^{-2t}}, where P(t)P(t) is number of population and tt is time in year. Let t1t_1 be the time when the population growth rate begin to decline and t2t_2 be the time when the population reach 8080 percent of its limit, then t1+t2t_1+t_2 can be expressed as Tln2T\ln 2. Determine the value of TT.

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