(3.) A major corporation is building a 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 Cove's population (in thousands) t years from now will be given by the Following function. 25t2 + 125t + 200 P(t) = t2 + 4t + 40 a. What is the current population (in number of people) of Glen Cove? b. What will be the population (in number of people) in the long run? The answer for part b is 25,000 people. 5. What was your procedure for part b? How do you know that was the appropriate procedure to use?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![### Population Growth in Glen Cove
#### Introduction
A major corporation is constructing a complex consisting of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As part of this development, planners have predicted that Glen Cove’s population (in thousands) \( t \) years from now will be represented by the following function:
\[ P(t) = \frac{25t^2 + 125t + 200}{t^2 + 4t + 40} \]
#### Questions
**(a) What is the current population (in number of people) of Glen Cove?**
**(b) What will be the population (in number of people) in the long run? The answer for part (b) is 25,000 people.**
#### Further Inquiry
**5. What was your procedure for part (b)? How do you know that was the appropriate procedure to use?**
#### Explanation and Solution Approach
For part (b), to determine the long-term population of Glen Cove, you need to analyze the function \( P(t) \) as \( t \) approaches infinity. The procedure typically involves finding the horizontal asymptote of the rational function. This can be done by comparing the degrees of the polynomial in the numerator and the denominator:
- If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
Given the function \( P(t) = \frac{25t^2 + 125t + 200}{t^2 + 4t + 40} \), the degree of the numerator and the denominator is 2. The leading coefficient of the numerator is 25, and the leading coefficient of the denominator is 1. Thus, the horizontal asymptote is:
\[ P(t) = \frac{25}{1} = 25 \]
Since the population is initially given in thousands, multiplying by 1,000 gives us the population:
\[ 25 \times 1,000 = 25,000 \]
This confirms that in the long run, the population of Glen Cove will be 25,000 people.
### Conclusion
Understanding and predicting population growth through functions helps in urban planning and managing resources effectively. This exercise demonstrates the use of mathematical functions to predict long-term population trends.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77461b06-0772-4165-878f-b81200c01932%2F4b86d6b3-d77d-4681-b1ab-f6738b892737%2Fuxyd82y.png&w=3840&q=75)
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