Paleontologist John Cisne has demonstrated that the survival of ancient manuscripts can be modeled by the logistic equation. For example, the number of copies of the Venerable Bede’s De Temporum Ratione was found to approach a limiting value over the five centuries after its publication in the year 725. Let G(t) represent the proportion of manuscripts known to exist after t centuries out of the limiting value, so that m = 1. Cisne found that for Venerable Bede’s De Temporum Ratione, k = 35 and G₀ = 0.00369.

a. Find the growth function G(t) for the proportion of copies of De Temporum Ratione found. Find the proportion of manuscripts and their rate of growth after the following number of centuries.

b. 1 

c. 2 

d. 3

e. What happens to the rate of growth over time?