Biologists stocked a lake with 160 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 4,000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years. (b) How long will it take for the population to increase to 2000 (half of the carrying capacity) ?

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Mathematics

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1 August, 00:13

Biologists stocked a lake with 160 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 4,000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years. (b) How long will it take for the population to increase to 2000 (half of the carrying capacity) ?

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Alisson Morse · Answer 1

The equation to find the number of fish after t years where y is the number of fish is:

y (t) = 160*3^ (t) (t < = 2.93 assuming that the maximum number of fish is 4000)

Therefore when y (t) = 2000 2000 = 160*3^ (t)

3^ (t) = 25/2 log 3 (3^ (t)) = log 3 (25/2) t = 2.29 years.