Answered: A company is considering to start…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section: Chapter Questions

Problem 18T

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Question

A company is considering to start harvesting salmon in a new region. They

model the salmon population size in that region by the equation

P′(t) = kP(t) − h,

with the natural growth rate of k = 2 salmon per year per population and a harvest rate h salmon per year. The company wants to decide how much salmon to harvest per year. The company estimates that currently there is 1mil salmon in that region.

harvest rate model
(i) h=500,000 ii) h=1mil, iii) h=2mil, iv) h=3mil

Is it a good idea to harvest salmon at the maximal sustainable rate projected by this model?

harve

Expert Solution

Step 1

$S i n c e, a c o m p a n y m o d e l s t h e s a l m o n p o p u l a t i o n s i z e i n a r e g i o n g i v e n b y t h e e q u a t i o n : - P' (t) = k P (t) - h - (i) S i n c e, i t i s g i v e n t h a t t h e n a t u r a l g r o w t h r a t e p e r y e r p e r p o p u l a t i o n i s 2, i . e, k = 2 . A l s o, c o m p a n y e s t i m a t e d t h a t c u r r e n t l y t h e r e i s 1 m i l s a l m o n i n t h a t r e g i o n . O u r A i m i s t o c a l c u l a t e h o w m u c h s a l m o n t o h a r v e s t p e r y e a r w i t h t h e g i v e n h a r v e s t r a t e m o d e l : - (i) h = 500, 000 = 0.5 m i l (i i) h = 1 m i l (i i i) h = 2 m i l a n d (i v) h = 3 m i l .$

Step 2

$S i n c e, f r o m e q u a t i o n (i), w e h a v e : - P' (t) = k P (t) - h \Rightarrow \frac{d P}{d t} - k P = - h - (i) I n t e g r a t i n g F a c t o r \Rightarrow e^{- \int k d t} = e^{- k t} S u p p o s e w e h a v e t h e d i f f e r e n t i a l \frac{d y}{d x} + P y = Q I t' s G e n e r a l S o l u t i o n i s g i v e n b y : - \Rightarrow y \times (I n t e g r a t i n g F a c t o r) = \int (Q \times I ntegrating Factor) d x + C \Rightarrow P \times (e^{- kt}) = - \int [h (e^{- kt})] d t \Rightarrow P \times (e^{- kt}) = \frac{- h}{- k} (e^{- kt}) + c \Rightarrow P \times (e^{- kt}) = \frac{h}{k} e^{- kt} + c D i v i d i n g b o t h s i d e s b y e^{- kt} w e h a v e : - \Rightarrow P = \frac{h}{k} + c (e^{- k t}) - (i i) A t t = 0, g i v e n k = 2, P = 1 m i l \Rightarrow 1 = \frac{h}{2} + c \Rightarrow c = 1 - \frac{h}{2} P u t t i n g t h e v a l u e o f c i n e q u a t i o n (i i), w e h a v e : - \Rightarrow P (t) = \frac{h}{2} + (1 - \frac{h}{2}) (e^{- 2 t}) - (i i i)$

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