Answer to Question #271770 in Calculus for Sara

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

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

a) $\frac{dy}{dt}=\frac{-l\cdot Ae^{-kt}(-k)}{(1+Ae^{-kt})^2}$

$=\frac{klAe^{-kt}}{(1+Ae^{-kt})^2}$

$y'(t=0)=\frac{kAl}{(1+A)^2}$

y is increasing at a rate of $\frac{kAl}{(1+A)^2}$

b) The rate of growth is always positive meaning y will increase with increase in t

c) $y''=\frac{(1+Ae^{-kt})^2(-k^2Ale^{-kt})-kAle^{-kt}\cdot 2(1+Ae^{-kt})(-kA)}{(1+Ae^{-kt})^4}=0$

$\implies(1+Ae^{-kt})(-1)+(+2)=0$

$\implies 1+Ae^{-kt}=2$

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

$t=\frac{ln{\frac{1}{A}}}{-k}$

Population is growing most rapidly at $t=\frac{ln{\frac{1}{A}}}{-k}$