Many environmental situations place effective limits on the growth of the number of an organism in an area. Many such limited-growth situations are described by the logistic function

where G₀ is the initial number present, M is the maximum possible size of the population, and k is a positive constant. The screens illustrate a typical logistic function calculation and graph.

Assume that G₀ = 100, M = 2500, k = 0.0004, and x = time in decades (10-yr periods).

(a) Use a calculator to graph the function, using 0 ≤ x ≤ 8 and 0 ≤ y ≤ 2500.

(b) Estimate the value of G(2) from the graph. Then evaluate G(2) algebraically to find the population after 20 yr.

(c) Find the x-coordinate of the intersection of the curve with the horizontal line y = 1000 to estimate the number of decades required for the population to reach 1000. Then solve G(x) = 1000 algebraically to obtain the exact value of x.

Transcribed Image Text:

MG, | G(x) = Go + (M— G)е кМк" FOAL PLOAT ITR E FAIAH H baA PLOAT TA EH AIAH 70 yFc/(1+ae( "bx) Logistic b=. 0953083462 a=9.233877705 c-61.88466971 (10. 20.30.40.50.60) :O_BLURRY