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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

The rate of growth of a fish population was modeled by the equation

G ( t ) = 60 , 000 e 0.6 t ( 1 + 5 e 0.6 t ) 2

where t is measured in years and G in kilograms per year. If the biomass was 25,000 kg in the year 2000, what is the predicted biomass for the year 2020?

To determine

To calculate: Thepredicted biomass for the year 2020.

Explanation

Given:

The rate of growth of a fish population is G(t)=60,000e0.6t(1+5e0.6t)2.

The biomass in the year 2000 is 25,000kg.

Calculation:

Calculate the net change from 2000 to 2020:

Netchangefrom 2000 to 2020=020G(t)dt

Substitute [60,000e0.6t(1+5e0.6t)2] for G(t).

020G(t)dt=020[60,000e0.6t(1+5e0.6t)2]dt (1)

The region lies between t=0 and t=20.

Take the consideration as follows:

u=1+5e0.6t (2)

Differentiate both sides of the Equation (2).

du=5e0.6(0.6)dte0.6dt=13du

Calculate the lower limit value of u using Equation (2).

Substitute 0 for t in Equation (2).

u=1+5e0.6(0)=6

Calculate the upper limit value of u using Equation (2).

Substitute 20 for t in Equation (2).

u=1+5e0.6(20)=1

Apply lower and upper limits for u in Equation (1).

Substituteu for (1+5e0

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