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

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