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) 25t^2+125t+200/t^2+5t+40

(1) What is the current population of Glen Road?

(2) What will be the population in the long run?

1) P (0) = 5000

2) P (t - -> ∞) = 25000

Step-by-step explanation:

P (t) = (25t² + 125t + 200) / (t² + 5t + 40)

1) Population at the moment corresponds to population at t = 0

P (0) = (25 (0²) + 125 (0) + 200) / (0² + 5 (0) + 40) = (0 + 0 + 200) / (0 + 0 + 40) = 5 thousand = 5000 (P was stated to be in thousands)

2) Population in the long term corresponds to the population as t - -> ∞

P (t) = (25t² + 125t + 200) / (t² + 5t + 40)

Divide through the numerator and denominator by t²

P (t) = (25 + (125/t) + (200/t²)) / (1 + 5/t + (40/t²))

P (t - -> ∞) = (25 + 0 + 0) / (1 + 0 + 0) (Since, (1/∞) = 0)

P (t - -> ∞) = 25 thousand = 25000