Answered: Given the logistic growth model: N(t)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Given the logistic growth model:

N(t) = k / 1 + be^-0.0065t

Find the optimum yield if b = 6 and the initial population is 689.

a) 4,823

b) 2,070

c) 98.43

d) 2411.5

e) 2398.11

Expert Solution

Step 1

Consider the given logistic growth model is $N (t) = \frac{k}{1 + b e^{- 0.0065 t}}$ .

When $b = 6$ , the above equation becomes $N (t) = \frac{k}{1 + 6 e^{- 0.0065 t}}$ .

The initial population is 689.

That is, when $t = 0$ , the value $N (t) = N (0) = 689$ .

Substituting the values $t = 0$ and $N (t) = 689$ in $N (t) = \frac{k}{1 + 6 e^{- 0.0065 t}}$ , we get

$689 = \frac{k}{1 + 6 e^{- 0.0065 \times 0}} \Rightarrow 689 = \frac{k}{1 + 6 e^{0}} \Rightarrow 689 = \frac{k}{7} \Rightarrow k = 689 \times 7 \Rightarrow k = 4823$

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