Question

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

Answer 1

Answer:

2.30 years

Step-by-step explanation:

The number of fish tripled in the first year, making a total of 240 * 3 = 720 fishes.

(a) The formula for logistic equation is as the following

$P = P_0e^{kt}$

where P0 = 240 is the number of fishes initially, we can plug in P = 720 and t = 1 to calculate the constant k

$720 = 240e^{1k}$

$e^k = 3$

$k = ln3 = 1.1$

b) Using the following formula

$P = P_0e^{kt}$

with P = 3000, P0 = 240, k = 1.1, we can calculate the number of years it takes to get to 3000 fishes

$3000 = 240e^{1.1t}$

$12.5 = e^{1.1t}$

$1.1t = ln12.5 = 2.53$

$t = 2.30 years$