Problem 25 - College Algebra

Question

For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010 . In \(2005,12,005\) , \(12,025\) people were afflicted. Find the linear function that models the number of people inflicted with the common cold, \(C,\) as a function of the year, \(t\)

Step-by-Step Solution

Answer

The linear model is \(C(t) = -205t + 12025\).

Step 1: Identify the Starting Point and Rate of Change

The initial number of people afflicted in the year 2005 is given as 12,025. It is also given that the number decreases by 205 each year. This will act as our rate of change.

Step 2: Set Up the Equation in Slope-Intercept Form

In a linear equation, the form used is \(C = mt + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope, \(m\), is -205 (as the number is decreasing by 205 each year), and the y-intercept, \(b\), is 12025 (the initial number of afflicted people in 2005).

Step 3: Convert the Year to a Variable, t

Define \(t\) such that \(t = 0\) corresponds to the year 2005. Thus, for any year \(t\), \(t = \text{year} - 2005\). This means in 2005, \(t=0\).

Step 4: Formulate the Linear Function

Substitute the values into the linear equation. Since each year is considered from 2005 onwards, the function is \[C(t) = -205t + 12025\]. Here, \(t\) is the number of years after 2005.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear function is crucial when modeling linear relationships. The equation is typically expressed as \( y = mx + b \). In this formula:

• \( y \) is the dependent variable (in this case, the number of people afflicted with a cold, \( C \)).
• \( m \) is the slope of the line, representing the rate of change.
• \( x \) is the independent variable, which can be a diverse input such as time.
• \( b \) is the y-intercept, or the starting value when \( x \) is zero.

In our scenario, the equation was modeled to understand how the number of afflicted people changes each year. The slope, or rate of change, \( m \), is \(-205\), indicating a yearly decrease, and the y-intercept, \( b \), is \( 12025 \), which was the number of people affected in 2005.

Rate of Change

The rate of change in a linear equation defines how the dependent variable alters as the independent variable increases or decreases. Here, the rate of change \( m \) is \(-205\). This negative number tells us:

• For each increase of 1 in the year variable, the number of people with colds decreases by 205.
• A negative slope indicates a decline, where a positive would show growth.

This linear relationship underscores the steady decline in the number of people with colds each year. By understanding the rate of change, we can make predictions or adjustments over time, knowing precisely how much to expect an attribute to decrease or increase per unit of the independent variable.

Year as a Variable

Representing years as a variable simplifies calculations and predictions. In this example, the year 2005 is a reference point, set as \( t = 0 \). The conversion is:

• For any given year, subtract 2005 from it to find \( t \).
• This makes 2006 correspond to \( t = 1 \), 2007 become \( t = 2 \), and so on.

This simplifies the function \( C(t) = -205t + 12025 \) by reducing it to simple arithmetic. By converting complex dates into manageable variables, calculations become straightforward, enabling easier data handling and analysis.

Modeling Real-World Scenarios

Modeling real-world scenarios with linear functions helps in forecasting and planning based on consistent data patterns. In our example, modeling how the number of people with colds changes over time allows:

• Authorities to anticipate healthcare needs.
• Researchers to analyze the effectiveness of interventions or environmental factors.
• Stakeholders to prepare for potential impacts on work or economic output.

By applying linear models, simplified insights are generated from observed consistent trends. Linear functions, like the one developed here, furnish a pragmatic approach to planning and responding to changing conditions. Mathematical modeling translates complex, raw data into clear, actionable information.