Question

The logistic equation models the growth of a population. P(t) = 4680 1 + 35e−0.4t

(a) Use the equation to find the value of k. k = 0.4 Correct: Your answer is correct.
(b) Use the equation to find the carrying capacity L. L = 4680 Correct: Your answer is correct.
(c) Use the equation to find the initial population
(d) Use the equation to determine when the population will reach 50% of its carrying capacity. (Round your answer to four decimal places.) years
(e) Use the equation to write a logistic differential equation that has the solution P(t). dP dt =

Answer 1

To find the initial population, substitute t = 0 into the given logistic equation. To determine when the population reaches 50% of its carrying capacity, substitute 0.5L for P(t) and solve for t using logarithms.

Step-by-step explanation:

(c) To find the initial population, we need to evaluate P(t) when t = 0.

Substituting t = 0 into the equation P(t) = 4680/(1 + 35e^-0.4t), we have P(0) = 4680/(1 + 35e^0). Since any number raised to the power of 0 is 1, we have P(0) = 4680/(1 + 35(1)).

Simplifying further, P(0) = 4680/(1 + 35) = 4680/36 = 130.

(d) To determine when the population will reach 50% of its carrying capacity, we need to find the value of t when P(t) = 0.5L.

Substituting 0.5L for P(t) in the equation P(t) = 4680/(1 + 35e^-0.4t), we have 0.5L = 4680/(1 + 35e^-0.4t).

Solving for t, we can use logarithms to rewrite the equation as ln(1 + 35e^-0.4t) = ln(4680/0.5L).

Then, we can isolate the variable t by using the properties of logarithms and evaluate it numerically.