Answer to Question #273777 in Calculus for mimi

Question

Q4. Suppose that a population yy grows according to the logistic model given by formula:

yy = LL

1 + AAee−kkkk .

a. At what rate is yy increasing at time tt = 0 ?

b. In words, describe how the rate of growth of yy varies with time.

c. At what time is the population growing most rapidly?

Answer 1

Solution;

Given;

$y=\frac{L}{1+Ae^{-kt}}$

(a)

Rate of growth;

$\frac{dy}{dt}=\frac{ALke^{-kt}}{(e^{-kt}+A)^2}$

At t=0;

$y'(0)=\frac{ALke^{0}}{(e^0+A)^2}=\frac{ALk}{(1+A)^2}$

(b)

The growth rate increase with time as as seen in (a).

(c)

The population is most rapid only when;

$1+Ae^{-kt}\neq0$

$Ae^{-kt}=-1$

$e^{-kt}=-\frac1A$

$-kt=-ln(\frac1A)$

$t=\frac1kln(\frac1A)$