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The Pacific halibut fishery has been modeled by the differential equation dy dt = ky 1 − y M 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 M = 9 × 107 kg, and k = 0.74 per year. (a) If y(0) = 2 × 107 kg, find the biomass a year later. (Round your answer to two decimal places.)

Answer :

gkosworldreign

Answer:

3.37 x 10^7 Kg

Step-by-step explanation:

Starting with the general solution equation for differential equation involving exponential population growth

[tex]y =\frac{c}{1 + Ae^{-kt} }[/tex]

c = 9 x 10^7Kg

k = 0.74 per year

[tex]y =\frac{9 * 10^{7} }{1 + Ae^{-0.74t} }[/tex]

[tex]A=\frac{c- y(0) }{y(0) } = \frac{9*10^{7} - 2*10^{7} }{2*10^{7}} \\\\A=3.5[/tex]

[tex]y =\frac{9 * 10^{7} }{1 + Ae^{-0.74t} } \\\\y =\frac{9 * 10^{7} }{1 + 3.5e^{-0.74t} } \\\\A year later , t= 1\\y =\frac{9 * 10^{7} }{1 + 3.5e^{-0.74*1} } = 33709144.04[/tex]

3.37 x 10^7 Kg

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