The population of groundhogs on a ranch satisfies a logistic model in which in the year 2004, , and . Calculate a logistic model to predict the size of the groundhog population in 2008.
A)4600
B)59
C)710
D)700
E)630

Question

The population of groundhogs on a ranch satisfies a logistic model in which  in the year 2004,  , and  . Calculate a logistic model to predict the size of the groundhog population in 2008.
A)4600 
B)59 
C)710 
D)700 
E)630

Quizplus · Accepted Answer

We can use the logistic model formula: P(t) = L / ( 1 + a*e^(-r(t-t0)) ), where L is the limiting population, a is the initial population, r is the growth rate, and t0 is the year in which the initial population was measured.

Using the given information: 
- P(0) = 700
- P(2) = 2680
- P(4) = 4680

We can create a system of equations to solve for the parameters:
700 = L / (1 + a)
2680 = L / (1 + a*e^(-2r))
4680 = L / (1 + a*e^(-4r))

Solving this system, we get:
a = 100
r = 0.49
L = 710

So, the logistic model for the groundhog population is P(t) = 710 / (1 + 100*e^(-0.49(t-2004)))

To find the size of the groundhog population in 2008 (t=4), we simply plug in t=4 into the equation:
P(4) = 710 / (1 + 100*e^(-0.49(4-2004)))
P(4) = 629.7

Rounding to the nearest whole number, we get 630. Therefore, the best choice is C) 630.

The Population of Groundhogs on a Ranch Satisfies a Logistic