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+2t+100
(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 in the population function p(t).
(a) When t=0, the population function becomes:
p(0) = 25(0)^2 + 125(0) + 200/(0)^2 + 2(0) + 100
Simplifying this expression:
p(0) = 0 + 0 + 200/0 + 0 + 100
However, since we cannot divide by zero, this expression is undefined. Therefore, we cannot determine the current population of Glen Cove using this equation.
(b) To find the population in the long run, we need to take the limit as t approaches infinity of the population function p(t).
Taking the limit as t approaches infinity:
lim t→∞ (25t^2 + 125t + 200)/(t^2 + 2t + 100)
Using the properties of limits, we can simplify this expression by dividing every term by the highest power of t in the numerator and denominator:
lim t→∞ [(25t^2/t^2) + (125t/t^2) + (200/t^2)] / [(t^2/t^2) + (2t/t^2) + (100/t^2)]
Simplifying further:
lim t→∞ (25 + 125/t + 200/t^2) / (1 + 2/t + 100/t^2)
As t approaches infinity, the terms with 1/t and 1/t^2 become negligible, and we are left with:
lim t→∞ (25 + 0 + 0) / (1 + 0 + 0)
Simplifying further:
lim t→∞ 25/1
Therefore, the population in the long run will converge to 25,000 people.
To calculate the current population of Glen Cove, we need to substitute the value of t as 0 into the given function p(t). This will give us the population at the present time.
(a) Current Population:
Substituting t = 0 into the function, we have:
p(0) = 25(0)^2 + 125(0) + 200 / (0)^2 + 2(0) + 100
p(0) = 0 + 0 + 200 / 0 + 0 + 100
p(0) = 200 / 100
p(0) = 2
Therefore, the current population of Glen Cove is 2 people.
To calculate the population in the long run, we need to consider what happens to the function as t approaches infinity. In this case, we can observe the behavior by looking at the leading terms of the function.
(b) Population in the Long Run:
As t approaches infinity, the leading terms in the numerator and denominator of the function dominate the expression. In this case, the leading terms are 25t^2 and t^2. Dividing these terms, we get:
25t^2 / t^2 = 25
Therefore, in the long run, the population of Glen Cove will approach 25 thousand people.
I have an inkling that your equation might be
p(t) = (25t^2 + 125t + 200)/(t^2 + 2t + 100)
current ----> t = 0
p(0) = 200/100 = 2, but it was measured in thousands
so the current population is 2000
in the long run ----> x --> ∞
as x gets bigger, we are left with the dominating terms of 25t^2/t^2 , thus the function approaches 25
so in the "long run" the population would be 25000