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## Homework Statement

The Pacific halibut fishery has been modeled by the differential equation.

[itex]\displaystyle\dfrac{dy}{dt}=ky\left(1-\dfrac{y}{M} \right)[/itex]

where

*y*(

*t*) is the biomass (the total mass of the members of the population) in kilograms at time

*t*(measured in years), the carrying capacity is estimated to be [itex]M = 7\times 10^7 kg[/itex], and [itex]k=0.78[/itex] per year.

(a) If [itex]y(0)= 2\times 10^7 kg[/itex], find the biomass a year later. (Round your answer to two decimal places.)

(b) How long will it take for the biomass to reach [itex]4\times 10^7 kg[/itex]? (Round your answer to two decimal places.)

## Homework Equations

[itex]\displaystyle P= \dfrac{K}{1+Ce^{-kt}}[/itex]

## The Attempt at a Solution

K = carrying capacity [itex]\implies 7\times 10^7 kg[/itex]

k = [itex]0.78[/itex] per year

At time 0, biomass is [itex]2\times 10^7 kg[/itex] [itex]\implies [/itex][itex]y(0)= 2\times 10^7 kg[/itex]

C = the difference between the carrying capacity and the initial capacity subtracted by 1.

[itex]C= \dfrac{7\times 10^7}{2 \times 10^7}-1=\dfrac{5}{2}[/itex]

[itex]P=\dfrac{7 \times 10^7}{1+\dfrac{5}{2}e^{-0.78\cdot1}}= 70000001.15 kg[/itex]

I'm trying to solve for the biomass (P) after 1 year. This answer doesn't seem correct. Am I using the wrong number for variable t? Or I'm not solving for P right away?