A major corporation is building a 4325 acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Clove's population (in thousands) t yr from now will be given by the following function.
P(t) = (45 t^2 + 125t + 100)/(t^2 + 6 t + 50)
(a) What is the current population of Glen Cove?
people
(b) What will be the population in the long run?
people
To find the current population of Glen Cove, we need to substitute t = 0 into the population function, P(t).
(a) Current population:
To find the current population, plug in t = 0 into the equation P(t) = (45t^2 + 125t + 100)/(t^2 + 6t + 50):
P(0) = (45(0)^2 + 125(0) + 100)/(0^2 + 6(0) + 50)
P(0) = 100/50
P(0) = 2
Therefore, the current population of Glen Cove is 2,000 people.
(b) Long run population:
In order to determine the long run population, we need to examine the behavior of the function P(t) as t approaches positive or negative infinity.
By observing the degrees of the numerator and denominator, we can determine that as t approaches positive or negative infinity, the population function can be simplified to:
P(t) ≈ (45t^2)/(t^2)
≈ 45
So, in the long run, the population of Glen Cove is estimated to stabilize at approximately 45,000 people.