A conservation organization releases 30 panthers into a preserve. After 3 years, there are 50 panthers in the preserve. The preserve has a carrying capacity of 150. Determine the time it Takes for the population to reach 110.
A)13.139 years
B)8.994 years
C)10.378 years
D)7.811 years
E)12.003 years

Question

Quizplus · Accepted Answer

To solve this problem, we can use the logistic growth model, which is given by:
P(t) = K / (1 + A * e^(-r*t))
where:
P(t) = population at time t
K = carrying capacity
A = (K - P(0)) / P(0)
P(0) = initial population
r = growth rate

We are given that P(0) = 30, K = 150, and P(3) = 50. We can use this information to solve for A and r:
A = (150 - 30) / 30 = 4
50 = 150 / (1 + 4 * e^(-r*3))
0.2 = 1 + 4 * e^(-r*3)
e^(-3r) = 0.2 / 4
-3r = ln(0.05)
r = -ln(0.05) / 3 = 0.561

Now we can use the logistic growth model to find the time it takes for the population to reach 110:
110 = 150 / (1 + 4 * e^(-0.561*t))
1 + 4 * e^(-0.561*t) = 150 / 110 = 1.364
4 * e^(-0.561*t) = 0.364
e^(-0.561*t) = 0.091
-0.561*t = ln(0.091)
t = -ln(0.091) / (-0.561) = 10.378 years

Therefore, the best choice is (C).

A Conservation Organization Releases 30 Panthers into a Preserve