Problem 75 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

AGING PoPULATION The population (in millions) of developed countries from 2005 through 2034 is projected to be $$ f(t)=3.567 t+175.2 \quad(5 \leq t \leq 35) $$ where \(t\) is measured in years. On the other hand, the population of underdeveloped/emerging countries over the same period is projected to be $$ g(t)=0.46 t^{2}+0.16 t+287.8 \quad(5 \leq t \leq 35) $$ a. What does the function \(D=g+f\) represent? b. Find \(D^{\prime}\) and \(D^{\prime}(10)\) and interpret your results.

Step-by-Step Solution

Answer

The function \(D(t) = 0.46t^2 + 3.727t + 463\) represents the total population (in millions) of both developed and underdeveloped/emerging countries combined from 2005 through 2034. The derivative \(D'(t) = 0.92t + 3.727\) represents the rate at which the total population is changing. The value \(D'(10) = 12.927\) means that in 2015, the total population (in millions) of both developed and underdeveloped/emerging countries was increasing at a rate of approximately 12.927 million people per year.

Step 1: Find function D (g+f) and its interpretation

To begin, we will find the function D by adding f(t) and g(t). Since these functions represent millions of people in developed and underdeveloped/emerging countries, respectively, D will represent the total population of both types of countries combined. So let's find the function D: $$ D(t) = f(t) + g(t) $$ Now let's add the given functions f(t) and g(t): $$ D(t) = (3.567t + 175.2) + (0.46t^2 + 0.16t + 287.8) $$ Simplify by combining like terms: $$ D(t) = 0.46t^2 + (3.567 + 0.16)t + (175.2 + 287.8) $$ $$ D(t) = 0.46t^2 + 3.727t + 463 $$ The function D(t) represents the total population (in millions) of both developed and underdeveloped/emerging countries, combined, from 2005 through 2034.

Step 2: Find the derivative D'(t)

Next, we need to find the derivative of D(t) with respect to t, which represents the rate at which the total population is changing. We'll denote the derivative as D'(t). To find D'(t), we will differentiate each term in D(t) with respect to t: $$ D'(t) = \frac{d}{dt}(0.46t^2) + \frac{d}{dt}(3.727t) + \frac{d}{dt}(463) $$ Using the power rule and constant rule for differentiation: $$ D'(t) = (2 \times 0.46)t^{2-1} + (3.727)t^{1-1} + 0 $$ Simplify the derivative: $$ D'(t) = 0.92t + 3.727 $$

Step 3: Find D'(10) and interpret the result

Now, we will find the value of D'(10), which represents the rate of change of the total population at t=10 years after 2005 (i.e., in 2015). Plug in t=10 into D'(t): $$ D'(10) = 0.92(10) + 3.727 $$ Evaluate the expression: $$ D'(10) = 9.2 + 3.727 = 12.927 $$ The result, D'(10) = 12.927, means that in 2015, the total population (in millions) of both developed and underdeveloped/emerging countries was increasing at a rate of approximately 12.927 million people per year.

Key Concepts

Population Growth

Population growth is a key concept when studying changes in the number of people living in certain areas over time. In the context of this exercise, we look at how populations in both developed and underdeveloped or emerging countries change.

• The function \( f(t) \) models the population growth in developed countries starting with a specific population and adjusting each year by a constant factor \( 3.567 \) million people per year.
• Meanwhile, \( g(t) \) captures the changing population in underdeveloped/emerging countries. Here, the growth isn't just a straight line but involves a quadratic component \( 0.46t^2 \). This component suggests that population changes might accelerate over time due to compound factors like birth rates outpacing death rates or immigration.

The combined population function \( D(t) \), which equals \( f(t) + g(t) \), essentially gives us the total population projection from 2005 to 2034 for both types of countries combined. Understanding this growth is crucial for planning resources and infrastructure for future societal needs.

Differentiation

Differentiation is a mathematical technique used to calculate the rate of change of a function. Here, understanding the rate of change of the population gives important insights.

- The derivative of the combined population function \( D(t) \), denoted as \( D'(t) \), represents how quickly the total population is changing at any time \( t \).
- By applying the power rule, which states that the derivative of \( t^n \) is \( nt^{n-1} \), we derive \( D'(t) = 0.92t + 3.727 \). This formula tells us that the growth in total population is not only constant but also affected by time \( t \).

Calculating \( D'(10) \) means we find out the specific rate at which the population was growing in 2015.

The derivative is particularly useful because it provides a snapshot of the population's growth trend at any moment, highlighting sharp increases or decreases in population sizes.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world phenomena. This exercise models the population growth through functions for both developed and underdeveloped countries.

• Each function is a simplification of potential population trends using assumptions based on historical data and projections.
• The linear function \( f(t) \) is suitable for fairly stable, predictable growth scenarios like many developed nations might experience.
• In contrast, \( g(t) \) with its quadratic term allows for capturing more complex growth patterns like those in underdeveloped regions experiencing rapid changes due to various social-economic factors.

Mathematical models like these help authorities and planners effectively allocate resources and make decisions regarding future needs. They offer a concrete way to explore scenarios, test hypotheses, and plan policies that adapt to expected population changes. The models aren't perfect but provide a basis for understanding and managing societal challenges related to population.