Answer to Question #272842 in Calculus for yobro

Question

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 increase of y at a time t=0;

$y'(t)=\frac{dy}{dt}=\frac{ALke^{-kt}}{(e^{-kt}+A)^2}$

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

(b)

The rate of growth as seen in (a) will always be positive meaning that y increases with time.

(c)

The population is most rapid only when;

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

Set;

$Ae^{-kt}=-1$

$e^{-kt}=\frac{-1}{A}$

$-kt=-ln(\frac{1}{A})$

$t=\frac{ln(\frac1A)}{k}$

Hence the population is most rapid at $t=\frac{ln(\frac1A)}{k}$