Question

Sunset Lake is stocked with 2700 rainbow trout and after 1 year the population has grown to 7650. Assuming logistic growth with a carrying capacity of 27000, find the growth constant k, and determine when the population will increase to 14500.

A. k = 0.3 yr⁽⁻¹⁾; yr⁽⁻¹⁾ = 2; The population will increase to 14500 after ≈ 3 years
B. k = 0.4 yr⁽⁻¹⁾; yr⁽⁻¹⁾ = 3; The population will increase to 14500 after ≈ 4 years
C. k = 0.5 yr⁽⁻¹⁾; yr⁽⁻¹⁾ = 4; The population will increase to 14500 after ≈ 5 years
D. k = 0.6 yr⁽⁻¹⁾; yr⁽⁻¹⁾ = 5; The population will increase to 14500 after ≈ 6 years

Answer 1

To find the growth constant k in the logistic growth model, substitute the given values into the formula and solve for k. Then, use the logistic growth equation to determine the time it takes for the population to reach 14500. The correct answer is Option C.

Step-by-step explanation:

To find the growth constant k in the logistic growth model, we can use the formula:

k = r(N/K)(1 - N/K)

Given that the initial population N₀ = 2700, the final population N = 7650, and the carrying capacity K = 27000, we can substitute these values into the formula and solve for k.

Once we have the value of k, we can use the logistic growth equation N(t) = (K / (1 + (K / N₀ - 1) * e-kt)) to determine when the population will increase to 14500 by solving for t.

Using the given options, we can calculate the value of k and determine the time it takes for the population to reach 14500. The correct answer is Option C. k = 0.5 yr⁽⁻¹⁾; yr⁽⁻¹⁾ = 4; The population will increase to 14500 after ≈ 5 years.