A conservation organization releases 20 wolves into a preserve. After 2 years, there are 35 wolves in the preserve. The preserve has a carrying capacity of 125. Determine the time it takes for the population to reach 80.
A) 9.020 years
B) 4.875 years
C) 6.259 years
D) 3.692 years
E) 7.884 years

Question

Quizplus · Accepted Answer

The growth of the wolf population can be modeled using the logistic growth equation, which is given by $ P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}} $, where $P(t)$ is the population at time $t$, $K$ is the carrying capacity, $P_0$ is the initial population, and $r$ is the growth rate. Given $K = 125$, $P_0 = 20$, and $P(2) = 35$, we first need to solve for $r$. Using $P(2) = 35$, we get $35 = \frac{125}{1 + \left(\frac{125 - 20}{20}\right)e^{-2r}}$. Solving this equation for $r$ gives us the growth rate. Once $r$ is known, we use the logistic growth equation to solve for $t$ when $P(t) = 80$, which gives us $t \approx 6.259$ years.

A Conservation Organization Releases 20 Wolves into a Preserve