Problem 91 - College Algebra and Calculus: An Applied Approach

Question

Median Age of U.S. Population The model \(A=15.68-0.037 t+6.131 \ln t, \quad 10 \leq t \leq 80\) approximates the median age \(A\) of the United States population from 1980 to \(2050 .\) In the model, \(t\) represents the year, with \(t=10\) corresponding to 1980 (see figure). (Source: U.S. Census Bureau)

Step-by-Step Solution

Answer

Following these steps should help in finding the median age of the U.S. population in any desired year between 1980 and 2050.

Step 1: Understand the model parameters

Start by understanding the variables in the model: \(A, t\). \(A\) represents the median age, while \(t\) represents the time in years. For the year 1980, \(t=10\), and \(t\) increases incrementally by 1 for each passing year. Therefore, to determine the median age of a specific year, assign the corresponding value to \(t\).

Step 2: Insert the 't' value in the model

Insert the appropriate \(t\) value in the model to represent the desired year. So, if you want to calculate the median age in 1990, assign \(t=20\) because this corresponds to 10 years post 1980. \(A = 15.68 - 0.037(20) + 6.131 \ln(20)\).

Step 3: Evaluate the model

Evaluate each term in the model separately and then collectively to get the value of \(A\). So, \(A = 15.68 - 0.74 + 6.131 \ln(20)\), then calculate the natural log of 20 and multiply it by 6.131, afterwards subtract 0.74 and add 15.68 to get the final value of the median age.

Key Concepts

Understanding Median Age Calculation

The median age is a statistical measure that defines the age that divides a population into two numerically equal groups. Half the people are younger and half are older. This is a critical measure as it gives us a middle point, offering insights into the overall age distribution of a population. Calculating the median age involves organizing the data and determining the middle point.
In the context of mathematical modeling, the median age can be calculated using functions that incorporate variables like time to predict changes over a period. For instance:

• Organize the ages in ascending order.
• If the number of observations is odd, the median is the middle number.
• If even, it's the average of the two middle numbers.

In mathematical models like those predicting demographic changes, parameters are used to automate the median age calculations over time.

Exploring Logarithmic Functions

Logarithmic functions are integral to understanding exponential growth and decay processes. These functions offer the practical advantage of simplifying complex calculations by converting multiplicative processes into additive ones. In mathematical modeling, logarithms are crucial when dealing with phenomena that increase or decrease at varying rates.
A logarithmic function has the form \(f(x) = a \ln(bx) + c\), where \(\ln\) represents the natural logarithm, commonly used in models relating to real-world data due to its natural exponential base \(e\). These functions are useful in:

• Transforming skewed data into more symmetrical data.
• Modeling phenomena such as human hearing or earthquake intensities.
• Facilitating calculations involving time or growth rates, as seen in demographic models.

Understanding and applying logarithmic functions in models can help predict changes over time more accurately.

The Role of Demographic Analysis in Mathematical Modeling

Demographic analysis involves studying populations to understand their size, structure, and distribution. This field uses various mathematical and statistical methodologies to interpret data and predict future demographic trends. By comprehensively analyzing population characteristics, demographic analysis provides valuable insights into the social and economic dynamics of populations.
In mathematical modeling, demographic analysis focuses on how variables such as age or employment rates change over time. This is especially useful for:

• Planning resources and services, like healthcare and education.
• Understanding population growth and its impact on the economy.
• Predicting future median ages and planning for aging populations.

Incorporating demographic analysis into mathematical models allows for more precise predictions and better decision-making in public policy and economic planning.