Problem 57 - College Algebra with Modeling and Visualization

Question

HIV Infections In 2006 there were 40 million people worldwide who had been infected with HIV. At that time the infection rate was 4.3 million people per year. (Source: United Nations AIDS and World Health Organization.) (a) Write a formula for a linear function \(f\) that models the total number of people in millions who were infected with HIV \(x\) years after 2006 (b) Estimate the number of people who may have been infected by the year 2012 .

Step-by-Step Solution

Answer

65.8 million people were estimated to be infected by 2012.

Step 1: Define the function

We know that in 2006, there were 40 million people infected and the infection rate is 4.3 million per year. The function can be written in the form \( f(x) = mx + b \), where \( m \) is the rate of change (slope) and \( b \) is the initial value (y-intercept). Here, \( m = 4.3 \) and \( b = 40 \).

Step 2: Write the linear function

Using the values from Step 1, substitute \( m \) and \( b \) into the linear function formula: \[ f(x) = 4.3x + 40 \]. This function models the number of people, in millions, infected with HIV \( x \) years after 2006.

Step 3: Calculate the years since 2006

To find \( x \) for the year 2012, we calculate the number of years after 2006: \( x = 2012 - 2006 = 6 \).

Step 4: Estimate the number of infections in 2012

Substitute \( x = 6 \) into the function \( f(x) = 4.3x + 40 \): \[ f(6) = 4.3(6) + 40 = 25.8 + 40 = 65.8 \]. So, approximately 65.8 million people are estimated to have been infected by 2012.

Key Concepts

Mathematical Modeling

Mathematical modeling is a powerful tool that helps us represent real-world scenarios using mathematical concepts and equations. In the context of our HIV infection problem, we used a linear model to predict the number of infected individuals over time. Why do we choose linear models for some situations? Because they provide a simple yet effective way to describe constant rates of change.
Using the problem as an example, HIV infections increase by a steady amount each year. Hence, by acknowledging this trend, the linear model \( f(x) = 4.3x + 40 \) represents this growth effectively. It breaks down complicated data into a form that is easier to understand and analyze, allowing us to make predictions and better decisions based on existing patterns.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. This form is expressed as \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept.

In our HIV problem, the line's slope \( m = 4.3 \) represents the rate of new infections per year. The y-intercept \( b = 40 \) indicates the initial number of infected people in 2006.
With the slope-intercept form, identifying the slope and y-intercept quickly helps understand the pattern and behavior of the data over time. It serves as an efficient and straightforward way to predict future outcomes or analyze trends in a dataset.

Rate of Change

The rate of change is a crucial concept in linear functions. It tells us how one quantity changes with respect to another. In our exercise, the rate of change (\( m \ = 4.3 \)) represents how the number of HIV infections increases every year. Understanding the rate, therefore, involves recognizing that for every additional year after 2006, 4.3 million more people are projected to be infected.
This constant rate gives us confidence in our predictions, assuming that the conditions remain similar over time. It enables a linear approach, simplifying our calculations and helping with the visualization and understanding of how the situation evolves.

Problem Solving in Mathematics

Effective problem solving in mathematics requires understanding the conditions and variables involved, and using appropriate mathematical tools to find a solution. For solving linear function problems like the one given, several key steps are involved:

• Identifying known variables and the change rate.
• Choosing the correct mathematical model or equation.
• Plugging in values to calculate unknowns.

For instance, to estimate HIV infections for 2012, we had to find the number of years between 2006 and 2012. Then, by substituting this into our model \( f(x) = 4.3x + 40 \), we successfully calculated the estimated number of infected people.
Problems in mathematics often require creativity, logical reasoning, and strategic thinking, all of which improve with practice and familiarity with different types of mathematical models.