Problem 94 - College Algebra with Modeling and Visualization

Question

Median Age The median age \(A\) in the United States during year \(x,\) where \(2000 \leq x \leq 2050,\) is projected to be $$ A(x)=0.07(x-2000)+35.3 $$ Use \(A(x)\) to estimate when the median age may reach 37 years. (Source: Bureau of the Census.)

Step-by-Step Solution

Answer

The median age will reach 37 years in 2024.

Step 1: Understand the Equation

The given function is \( A(x) = 0.07(x - 2000) + 35.3 \). It represents the median age \( A \) as a function of year \( x \).

Step 2: Set the Equation for 37 Years

To find out when the median age \( A(x) \) is 37 years, set \( A(x) = 37 \). So, the equation is \[ 37 = 0.07(x - 2000) + 35.3 \].

Step 3: Subtract Constant Term

Rearrange the equation to start isolating \( x \). Subtract 35.3 from both sides: \[ 37 - 35.3 = 0.07(x - 2000) \]. This simplifies to \[ 1.7 = 0.07(x - 2000) \].

Step 4: Solve for x

Solve for \( x \) by first dividing both sides by 0.07: \[ x - 2000 = \frac{1.7}{0.07} \]. Simplifying gives \[ x - 2000 = 24.29 \]. Add 2000 to both sides to find \( x \): \[ x = 2000 + 24.29 = 2024.29 \].

Step 5: Interpret the Result

Since year \( x \) must be a whole number, we round \( 2024.29 \) to the nearest year. The median age is projected to reach 37 years in the year 2024.

Key Concepts

Linear Equation

In the world of algebraic functions, a linear equation is an expression that represents a straight line when plotted on a graph. These equations are typically written in the standard form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. In the exercise, the function \( A(x) = 0.07(x - 2000) + 35.3 \) is a linear equation.
The 'slope', \( m = 0.07 \), tells us the rate at which the median age changes for each year increase. Meanwhile, \( 35.3 \) is where the line intersects the y-axis, representing the median age in the year 2000. By understanding this structure, we can predict values, as illustrated by determining when the median age reaches 37 years.

• The value of \( m \): Represents the rate of change.
• \( c \): The starting value (y-intercept) of the function.

Recognizing a function as linear, like \( A(x) \), allows for easy manipulation to solve real-world problems by setting values and isolating variables.

Problem Solving

Problem solving involves a series of logical steps to find solutions. The exercise about median age estimation follows this strategy, starting with understanding the function's representation.
The goal was to find out in which year the median age reaches 37. The first step is to set the equation equal to 37, allowing us to isolate and solve for \( x \), which stands for the year.

• Set the equation to the desired outcome: \( A(x) = 37 \).
• Simplify the equation by isolating terms: Subtract constants and divide to isolate for \( x \).
• Find the resulting year by performing arithmetic operations.

The problem-solving approach efficiently transforms a seemingly complex task into manageable steps, leading to an achievable solution.

Mathematical Modeling

Mathematical modeling uses mathematical expressions to represent, understand, and predict real-world scenarios. The given exercise uses a linear equation to model the projection of median age over time.
This model, represented by \( A(x) \), allows us to estimate the median age for future years, giving clear insights into demographic changes over time. By setting parameters and using a defined rate of change (slope), models like these help policymakers and researchers anticipate societal needs.
In general, mathematical modeling involves:

• Identifying the problem or scenario to model (e.g., median age increase).
• Creating mathematical representations to describe the relationships (e.g., linear equations).
• Using the model to predict future outcomes and aid decision-making.

This form of modeling bridges abstract mathematics with practical applications, offering valuable tools for foresight and planning.