Problem 88 - A Graphical Approach to College Algebra

Question

U.S. HIV/AIDS Infections In 2010 , there were approximately 1.2 million people in the United States living with HIV/AIDS. At that time the infection rate was \(50,000\) people per year. (a) Find values for \(m\) and \(b\) so that \(y=m x+b\) models the total number of people \(y\) in millions who were living with HIV/AIDS \(x\) years after 2010 . (b) Find \(y\) for the year 2014 . Interpret your result.

Step-by-Step Solution

Answer

(a) \( m = 0.05 \), \( b = 1.2 \); (b) 1.4 million in 2014.

Step 1: Understand the Linear Model

The problem specifies a linear relationship in the form of the equation \( y = mx + b \), where \( y \) is the number of people (in millions) with HIV/AIDS, \( x \) is the number of years after 2010, \( m \) is the rate of increase (slope) per year, and \( b \) is the initial number of people in 2010.

Step 2: Identify the Given Values to Set Up the Equation

From the problem statement, we know the initial value \( b = 1.2 \) million (since there were 1.2 million people in 2010). The rate of infection \( m \) is \( 50,000 \) people per year, which is equivalent to \( 0.05 \) million per year.

Step 3: Write the Equation for the Linear Model

With \( m = 0.05 \) and \( b = 1.2 \), the equation becomes:\[ y = 0.05x + 1.2 \]

Step 4: Evaluate the Equation for 2014

To find \( y \) for the year 2014, calculate the number of years after 2010, which is \( x = 2014 - 2010 = 4 \). Substitute \( x = 4 \) into the linear equation:\[ y = 0.05(4) + 1.2 \]

Step 5: Perform the Calculation

Calculate the expression:\[ y = 0.2 + 1.2 = 1.4 \]Thus, in 2014, there were 1.4 million people living with HIV/AIDS.

Step 6: Interpretation of the Result

This indicates that in 2014, the total number of people living with HIV/AIDS had increased to 1.4 million from 1.2 million in 2010, reflecting the consistent infection rate.

Key Concepts

Slope and Intercept

In the context of linear equations, the terms **slope** and **intercept** play vital roles in defining the relationship between two variables. The slope is represented by the letter \( m \) and indicates how much \( y \) changes for every unit increase in \( x \).
Think of it as the 'steepness' of the line in the graph of the equation.On the other hand, the intercept, denoted by \( b \), is the starting point of the line on the \( y \)-axis.
It tells us the value of \( y \) when \( x \) is zero. In our exercise, \( b = 1.2 \) million, showing that initially, 1.2 million people were living with HIV/AIDS in 2010.Key Points:

• **Slope (m):** Represents the change in \( y \) per unit change in \( x \).
• **Intercept (b):** Initial value of \( y \) when \( x = 0 \).
• The formula \( y = mx + b \) combines both aspects to model a linear trend.
• Our slope of \( 0.05 \) million shows how the HIV/AIDS cases increased annually.

Rate of Change

The term "rate of change" describes how a quantity changes in relation to another.
In our linear model, it is intuitive to connect this to the concept of slope. The rate indicates how quickly or slowly something happens.
It's measured as the ratio of the change in one variable relative to the change in another. For example, in this problem, the rate of change is equivalent to 50,000 people per year, converting to 0.05 million annually.
This means for each year that passes after 2010, 50,000 more people are living with HIV/AIDS. Understanding Rate of Change:

• The rate of 0.05 means an increase of 50,000 people each year.
• It highlights the steady and consistent growth observed in the dataset.
• Knowing the rate helps in predicting future values effectively.

By identifying rates, we can model significant trends and plan appropriate measures effectively.

Years After 2010

The "Years After 2010" concept serves a crucial role in this problem by defining our timeline.
In mathematics, setting a baseline year helps make calculations more straightforward. Here, the number of years after 2010 is represented by \( x \) in our linear equation.For instance, to calculate values like those provided for the year 2014, the formula takes "years after 2010" into account by subtracting 2010 from 2014 to get 4.
This time frame helps in finding \( y \), the predicted number of people with HIV/AIDS.Why This Matters:

• It gives a reference point (2010 was the start of the data).
• Translates real-world years into a mathematical variable \( x \).
• Simplifies calculations for any given year after 2010.
• Allows for predictive modeling of data trends to understand future changes.

By clearly defining the parameter "years after 2010," we can ensure the equation remains relevant and easily applicable for future predictions.