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 + 300)/(t^2 + 4 t + 150)
(a) What is the current population of Glen Cove?
people
(b) What will be the population in the long run?
people
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 + 300)/(t^2 + 4 t + 150)
(a) What is the current population of Glen Cove?
people
(b) What will be the population in the long run?
people
No one has answered this question yet.
To find the current population of Glen Cove, we need to substitute t = 0 into the function P(t).
(a) Current population:
P(0) = (45(0)^2 + 125(0) + 300) / ((0)^2 + 4(0) + 150)
= (0 + 0 + 300) / (0 + 0 + 150)
= 300 / 150
= 2
Therefore, the current population of Glen Cove is 2,000 people (since the function is given in thousands).
To find the population in the long run, we need to analyze the behavior of the function as t approaches infinity.
(b) Long-run population:
As t approaches infinity, the highest power term (t^2) dominates the function. Therefore, we can ignore the lower degree terms (the constants and t coefficient) in the numerator and denominator.
P(t) ≈ (45 t^2) / (t^2)
≈ 45
Thus, in the long run, the population of Glen Cove would be approximately 45,000 people.