Answer:
the logistic function that models the population of this species as a function of time is P(t) = 7000 / (1 + 0.05 * e^(-0.1682 * t)).
Explanation:
We can use the logistic growth equation to model the population of this species as a function of time:
P(t) = K / (1 + A * e^(-r * t))
Where:
P(t) is the population at time t
K is the carrying capacity of the environment
A is the initial population as a proportion of the carrying capacity (A = P(0)/K)
r is the growth rate of the population
We are given that the initial population A is 350/7000 = 0.05 (since the carrying capacity is 7000). We are also given that after 5 years the population has grown to 900. So we can use this information to find the growth rate r:
P(5) = 900 = K / (1 + A * e^(-r * 5))
We also know that the carrying capacity K is 7000. Substituting these values, we get:
900 = 7000 / (1 + 0.05 * e^(-r * 5))
Multiplying both sides by the denominator and simplifying, we get:
1 + 0.05 * e^(-r * 5) = 7.777...
Subtracting 1 from both sides, we get:
0.05 * e^(-r * 5) = 6.777...
Dividing both sides by 0.05, we get:
e^(-r * 5) = 135.555...
Taking the natural logarithm of both sides, we get:
-ln(135.555...) = -r * 5
Solving for r, we get:
r = 0.1682...
Now that we have the growth rate r, we can use the logistic growth equation to find the function that models the population as a function of time:
P(t) = 7000 / (1 + 0.05 * e^(-0.1682 * t))
Therefore, the logistic function that models the population of this species as a function of time is P(t) = 7000 / (1 + 0.05 * e^(-0.1682 * t)).