SOLUTION: A lake in Northern Ontario has recovered from an acid spill that killed all of its trout. A restocking program puts 800 trout in the lake. Ten years later, the population is estima

Question 1173914: A lake in Northern Ontario has recovered from an acid spill that killed all of its trout. A restocking program puts 800 trout in the lake. Ten years later, the population is estimated to be 6000. The carrying capacity of the lake is believed to be 8000. Consider a logistic model of the form P(t)= c/1+a(b)^t to model the growth of the trout population, where P(t) is the number of trout, t is the number of years, and a, b, and c are the parameters. After restocking, how many years will it take for the number of trout to reach 2349?
Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!
As t gets very large, the population tops out at 8000. In order for this to
happen, b must be less than 1, so that P(infinity) does not go down to zero.
For t large, b^t goes to zero, so P(inf) = c/(1+0) = 8000 -> c = 8000
At t=0, P(0) = 800 = 8000/(1 + a*b^0) = 8000/(1+a) -> a = 9
P(10) = 6000 = 8000/(1 + 9*b^10) -> b = 0.71922
Thus the equation for the population model is P(t) = 8000/(1+9(0.71922)^t)
2349 = 8000/(1+9(0.71922)^t) -> solve for t -> 0.71922^t = (1/9)(8000/2349 - 1)
tlog(0.71922) = log((1/9)(8000/2349 - 1) -> t = 4 years
The graph is below:

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