How many black-nosed rabbits will be in the range 5 years from 1990?
Exponential Law for Rabbit Population
The exponential law states that the growth of the black-nosed rabbit population in a national forest can be modeled using the formula Q(t) = Q(0)e^(kt), where Q(t) represents the population at time t, Q(0) is the initial population, k is the growth rate, and e is the base of the natural logarithm (approximately equal to 2.71828).
Calculation for Rabbit Population
Given that conservationists tagged 90 black-nosed rabbits in a national forest in 1990 and then tagged 180 rabbits in the same range in 1993, we can determine the growth rate using the exponential law formula.
Let's calculate the growth rate using the provided data:
180 = 90 * e^(3k)
2 = e^(3k) (obtain the natural log on both sides)
ln(2) = 3k
k â 0.23104906
Therefore, the growth rate k is approximately 0.23104906. We can now calculate the rabbit population in the range 5 years from 1990 using the formula Q(5) = 90 * e^(0.23104906 * 5).
Population Prediction
By substituting the values into the formula, we get:
Q(5) = 90 * e^(0.23104906 * 5) â 285.73
Therefore, we can conclude that approximately 285 black-nosed rabbits will be in the range 5 years from 1990 based on the exponential growth model.