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2 months ago

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Biologists stocked a lake with 240 fish and estimated the carrying capacity (the maximal population for the fish of that species

in that lake) to be 6,000. 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.

P =

(b) How long will it take for the population to increase to 3000? (Round your answer to two decimal places.)

=yr

1 answer:

Inessa [12.5K]2 months ago

6 0

Answer:

2.30 years

Step-by-step explanation:

The fish population increased threefold in the first year, resulting in 240 * 3 = 720 fish.

(a) The logistic equation can be represented as follows

$P = P_0e^{kt}$

where P0 = 240 denotes the initial fish count; substituting P = 720 and t = 1 allows us to determine the constant k

$720 = 240e^{1k}$

$e^k = 3$

$k = ln3 = 1.1$

b) By applying the formula below

$P = P_0e^{kt}$

with P set to 3000, P0 as 240, and k equal to 1.1, we can compute the duration needed for the population to reach 3000 fish

$3000 = 240e^{1.1t}$

$12.5 = e^{1.1t}$

$1.1t = ln12.5 = 2.53$

$t = 2.30 years$

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