The logistic growth function P(t)= 160/ 1 + 9e^-0.165t describes the population of the endangered species of elk "t" years after they were introduced into the habitat.
a. How many elk were initially introduced into the habitat?
b. How many elk are expected after 6 years?
What is the limited size of elk population sustained by the habitat?
P(0) = 160/(1+9) = 16
Just plug in t=6 for the population after 6 years.
Note that as t->∞, e^-0.165t -> 0
So, the maximum possible population is 160
See
http://www.wolframalpha.com/input/?i=160%2F+%281+%2B+9e^%28-0.165t%29%29+for+t+%3D+0..50
The population of a southern city follows the exponential law. If the population doubled in size over 29 months and the current population is 30,000, what will the population be 33 years from now
To answer the questions, let's break down the problem step by step:
a. How many elk were initially introduced into the habitat?
The logistic growth function P(t) describes the population of the elk "t" years after they were introduced into the habitat. To find the initial population, we need to find the value of P(0).
Substituting t = 0 into the given equation:
P(0) = 160 / (1 + 9e^(-0.165*0))
Simplifying:
P(0) = 160 / (1 + 9e^0)
Since e^0 equals 1:
P(0) = 160 / (1 + 9*1)
P(0) = 160 / (1 + 9)
P(0) = 160 / 10 = 16
Therefore, 16 elk were initially introduced into the habitat.
b. How many elk are expected after 6 years?
To find the number of elk expected after 6 years, we need to calculate P(6).
Substituting t = 6 into the given equation:
P(6) = 160 / (1 + 9e^(-0.165*6))
Simplifying:
P(6) = 160 / (1 + 9e^(-0.99))
Since e^(-0.99) is a numeric value, we can use a calculator to evaluate it:
P(6) ≈ 160 / (1 + 9 * 0.372) ≈ 160 / (1 + 3.348) ≈ 160 / 4.348
P(6) ≈ 36.78
Approximately 36.78 elk are expected after 6 years.
c. What is the limited size of the elk population sustained by the habitat?
The limited size of the population sustained by the habitat is the maximum population that the habitat can support. In logistic growth, this is called the carrying capacity.
Since the carrying capacity is not provided directly in the equation, we need some additional information to determine it, such as the maximum value of P(t) as t approaches infinity or any specific information about the habitat's carrying capacity.
Without that additional information, we cannot determine the limited size of the elk population sustained by the habitat.
To answer these questions, we need to understand the logistic growth function and how to interpret it.
The logistic growth function is given by:
P(t) = (K/(1 + ae^(-bt)))
Where:
P(t) represents the population at time t
K is the carrying capacity, which represents the maximum population size that the habitat can sustain
a is the initial population size when t = 0
b is the growth rate parameter
Now let's answer the questions:
a. How many elk were initially introduced into the habitat?
To determine the initial population size (a), we can substitute the given values into the logistic growth function.
P(t) = 160 / (1 + 9e^(-0.165 * t))
Since we're looking for the initial population size, we need to find the population at time t = 0.
Plugging in t = 0, we have:
P(0) = 160 / (1 + 9e^(-0.165 * 0))
= 160 / (1 + 9e^0)
= 160 / (1 + 9)
= 160 / 10
= 16
Therefore, the initially introduced population of elk is 16.
b. How many elk are expected after 6 years?
To find the population after 6 years, we substitute t = 6 into the logistic growth function.
P(t) = 160 / (1 + 9e^(-0.165 * t))
P(6) = 160 / (1 + 9e^(-0.165 * 6))
Using a calculator, we can evaluate the expression:
P(6) ≈ 160 / (1 + 9e^(-0.165 * 6))
≈ 160 / (1 + 9e^(-0.99))
≈ 160 / (1 + 9 * 0.3713)
≈ 160 / (1 + 3.342)
≈ 160 / 4.342
≈ 36.8
Therefore, there are approximately 36 elk expected after 6 years.
c. What is the limited size of the elk population sustained by the habitat?
The carrying capacity (K) represents the maximum population size that the habitat can sustain. It is a limiting factor that prevents the population from growing indefinitely.
To find the carrying capacity (K), we can look at the long-term behavior of the logistic growth function. As t approaches infinity, the term ae^(-bt) approaches 0, making the denominator approach 1.
So, when t approaches infinity, the logistic growth function becomes:
P(t) = K / 1
= K
Therefore, the limited size of the elk population sustained by the habitat is represented by the carrying capacity (K) in the logistic growth function. In this case, we don't have a specific value for K provided in the given function, so it cannot be determined without additional information.