Answer to Question #273777 in Calculus for mimi

Question #273777

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?


1
Expert's answer
2021-12-01T12:33:42-0500

Solution;

Given;

y=L1+Aekty=\frac{L}{1+Ae^{-kt}}

(a)

Rate of growth;

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

At t=0;

y(0)=ALke0(e0+A)2=ALk(1+A)2y'(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+Aekt01+Ae^{-kt}\neq0

Aekt=1Ae^{-kt}=-1

ekt=1Ae^{-kt}=-\frac1A

kt=ln(1A)-kt=-ln(\frac1A)

t=1kln(1A)t=\frac1kln(\frac1A)



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