Question

A major corporation is building a 4325-acre complex of homes, offices,stores, schools, and churches in the rural community of Glen Road. As a result of this development, the planners have estimated that Glen Road's population (in thousands) t years now will be given.

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?

Answer 1

a) Current population of Glen Road = 50,000

b) Population of Glen Road in the long run = 25,000

Explanation:

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

where P is population (in thousands) t years from now.

a) The current population of Glen Road

At the current moment, t = 0

P(0) = (25(0²) + 125(0) + 200)/((0²) + 5(0) + 40)

P(0) = (0+0+200)/(0+0+40)

P(0) = (200/40) = 50,000

b) The population in the long run

In the long run, t --> ∞

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

Divide numerator and denominator by t²

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

As t --> ∞

P(t --> ∞) = (25 + (125/∞) + (200/∞))/(1 + (5/∞) + (40/∞))

Every number divided by infinity goes to 0.

P(t --> ∞) = (25 + 0 + 0)/(1 + 0 + 0)

P(t --> ∞) = (25/1)

P(t --> ∞) = 25,000