Question

Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population follows a logistic growth model:

P(t) =

d

1 + ke−ct

where c, d, and k are positive constants. For a certain fish population in a small pond d = 1400, k = 13, c = 0.2, and t is measured in years. The fish were introduced into the pond at time t = 0.

(a) How many fish were originally put in the pond?


Correct: Your answer is correct.

fish


(b) Find the population after 10, 20, and 30 years. (Round your answers to the nearest whole number.)

10 years Fish

20 years fish

30 years fish

Answer 1

Answer:

a) 19,600 fishes were originally put in the pond.

b) Population of fishes after 10 years = 3,863

Population of fishes after 20 years = 1,733

Population of fishes after 30 years = 1,403

Step-by-step explanation:

The population follows a logistic model

P(t) = d (1 + ke⁻ᶜᵗ)

For a fish pond,

d = 1400, k = 13, c = 0.2

Inserting the values of these constants

P(t) = 1400 (1 + 13 e⁻⁰•²ᵗ)

a) How many fish were originally put in the pond?

At the start of the whole thing, t = 0

P(t=0) = 1400 (1 + 13 e⁰) = 1400 × 14 = 19,600

Hence, 19,600 fishes were originally put in the pond.

b) Find the population after 10, 20, and 30 years.

P(t) = 1400 (1 + 13 e⁻⁰•²ᵗ)

At t = 10, 0.2t = 0.2 × 10 = 2

P(t=10) = 1400 (1 + 13e⁻²) = 1400 (1 + 1.759) = 3,863.1 = 3,863

At t = 20, 0.2t = 0.2 × 20 = 4

P(t=20) = 1400 (1 + 13e⁻⁴) = 1400 (1 + 0.238) = 1,733.3 = 1,733

At t = 30, 0.2t = 0.2 × 30 = 6

P(t = 30) = 1400 (1 + 13e⁻⁶) = 1400 (1 + 0.00248) = 1,403.47 = 1,403

Hope this Helps!!