Problem 28 - 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. If the function \(C\) is graphed, find and interpret the slope of the function.

Step-by-Step Solution

Answer

The slope is -205, meaning 205 fewer people are afflicted each year.

Step 1: Identify the Initial Data

The problem states that in the year 2005, there were 12,025 people afflicted with the common cold. This will be our starting data point: Year = 2005, Afflicted People = 12,025.

Step 2: Identify the Rate of Change

The number of people afflicted decreases by 205 every year. This means the rate of change, or the slope, of the function is -205.

Step 3: Understand the Slope-Intercept Form

In the slope-intercept form of a linear function, which is \( y = mx + b \), \( m \) represents the slope. In our context, \( m = -205 \), indicating the number of afflicted people decreases by 205 each year.

Step 4: Interpret the Slope

The slope of the function represents the annual change in the number of people afflicted with the common cold. Specifically, a slope of -205 indicates a decrease of 205 people per year from 2005 to 2010.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most popular ways to express linear equations in mathematics. This form makes it easier to understand and graph linear functions. A linear equation in slope-intercept form can be written as \( y = mx + b \). Here, \( m \) stands for the slope of the line, which indicates how steep the line is and the direction it moves. The \( b \) represents the y-intercept, the point where the line crosses the y-axis.
This form is extremely helpful as it directly gives us two important pieces of information:

• The slope \( m \), which shows the rate of change.
• The y-intercept \( b \), which provides a starting point at \( x = 0 \).

Rate of Change

The rate of change is a crucial concept when dealing with linear functions. In our scenario, the rate of change tells us how the number of people affected by the common cold changes over time. Specifically, it reflects how this number decreases or increases year by year.
In our case, the rate of change is -205, which indicates a consistent decrease in the number of affected people each year from 2005 to 2010. The minus sign signifies that it is going down rather than up.
Understanding the rate of change helps a lot because:

• It shows the trend of decreasing numbers each year.
• It helps predict future values by indicating how much less each successive year's value will be.

Interpretation of Slope

The slope of a line in a graph represents how one variable changes in relation to another. In the context of the common cold problem, the slope is crucial for understanding how the number of afflicted people changes as time progresses.
In our exercise, the slope is -205. This means that for every year, the number of people afflicted with the common cold decreases by 205. Thus, the slope provides insight into:

• The negative sign indicates that the trend is decreasing over time.
• The actual number, 205, shows the quantity of change each year.

This interpretation of slope lets us understand not just how much things are changing, but also the nature of those changes—whether they are increasing or decreasing, and by how much per unit of time or another measure. This insight is extremely powerful for making predictions and understanding trends in data.