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Sustainable Fish Population

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A fish population in Lake Pisces in which fishing is prohibited increases according to the standard logistic model $R = 2F - 0.01F^{2}$, where $R$ is the rate at which the fish population increases, as m​easured in fish per month, and $F$ is the fish population in any given month.

In the standard logistic model, the fish population does not have unlimited growth. Eventually, the growth rate approaches zero when the fish population approaches the maximum sustainable level for the natural ecosystem of the lake.

Lake Poisson is a similar lake, except that fishing is permitted. Although the same logistic model describes the natural growth of the fish population, fish are also caught and removed from the lake at a constant rate of 75 fish per month.

What is the maximum sustainable fish population in Lake Pisces?

What is the minimum number of fish required to be in Lake Poisson in any given month to avoid the collapse of the fish population due to overfishing?

Sustainable population in Lake Pisces

Minimum population in Lake Poisson

Sustainable population in Lake Pisces

Minimum population in Lake Poisson

25

Sustainable population in Lake Pisces

Minimum population in Lake Poisson

50

Sustainable population in Lake Pisces

Minimum population in Lake Poisson

75

Sustainable population in Lake Pisces

Minimum population in Lake Poisson

94

Sustainable population in Lake Pisces

Minimum population in Lake Poisson

100

Sustainable population in Lake Pisces

Minimum population in Lake Poisson

200