Problem 79
Question
Find a linear equation in slope-intercept form that models the given description. Describe what each variable in your model represents. Then use the model to make a prediction. In \(1995,60 \%\) of U.S. adults read a newspaper and this percentage has decreased at a rate of \(0.7 \%\) per year since then.
Step-by-Step Solution
Verified Answer
The linear model is \(y = -0.7x + 60\). 'y' represents the percentage of U.S. adults who read a newspaper, 'x' represents the number of years since 1995, '-0.7x' is the decrease percentage each year, and '60' is the initial percentage in 1995. For example, it's predicted that \(53\%\) of U.S. adults read a newspaper in the year 2005.
1Step 1: Identify the Slope and Intercept
First, determine the slope, which describes the rate of decrease per year. Given in the problem, the percentage of adults reading a newspaper has decreased at a rate of \(0.7\%\) per year, thus the slope (m) is \(-0.7\). The initial value, or the y-intercept, is the percentage of U.S adults that read a newspaper in 1995, which is \(60\%\). Thus, the y-intercept (b) is \(60\).
2Step 2: Write the Equation
Next, insert the slope and y-intercept identified in the first step into the linear equation in slope-intercept form \(y = mx + b\). This gives \(y = -0.7x + 60\). In this equation, 'y' represents the percentage of U.S. adults who read a newspaper, 'x' represents the number of years since 1995, '-0.7x' is the decrease in the percentage of adults reading a newspaper each year, and '60' is the initial percentage of adults who read a newspaper in 1995.
3Step 3: Make Predictions
To use the equation to make predictions, simply input the desired number of years since 1995 into the equation. For instance, to predict the percentage of U.S adults reading a newspaper 10 years after 1995, we have: \(y = -0.7 \times 10 + 60 = 53\%\). So, it is predicted that \(53\%\) of U.S adults read the newspaper in the year 2005.
Key Concepts
Understanding the Rate of ChangeInterpreting the Y-InterceptMaking Linear Model Predictions
Understanding the Rate of Change
The rate of change in a linear equation is a crucial concept that indicates how much one variable changes in response to a change in another variable. In the context of the provided exercise, the rate of change is represented by the slope of the line and is specifically a measure of the decline in newspaper readership among U.S. adults over time.
In the exercise, we are given that the rate of change is -0.7% per year. This means that each year, the percentage of adults reading a newspaper decreases by 0.7%. Mathematically, when expressed in the slope-intercept form of a linear equation, which is \(y = mx + b\), the 'm' stands for the slope (-0.7 in this case).
Understanding the slope as a rate of change helps in making sense of how variables are related in a linear equation. If the slope is negative, as it is here, it indicates a decrease. Conversely, a positive value would suggest an increase. Therefore, being able to interpret the rate of change is fundamental not only to solving linear equations but also to understanding the trends and relationships they represent.
In the exercise, we are given that the rate of change is -0.7% per year. This means that each year, the percentage of adults reading a newspaper decreases by 0.7%. Mathematically, when expressed in the slope-intercept form of a linear equation, which is \(y = mx + b\), the 'm' stands for the slope (-0.7 in this case).
Understanding the slope as a rate of change helps in making sense of how variables are related in a linear equation. If the slope is negative, as it is here, it indicates a decrease. Conversely, a positive value would suggest an increase. Therefore, being able to interpret the rate of change is fundamental not only to solving linear equations but also to understanding the trends and relationships they represent.
Interpreting the Y-Intercept
The y-intercept in a linear equation is the value at which the line crosses the y-axis. It is the value of the dependent variable when the independent variable is zero.
The initial readership percentage in our exercise, 60%, is the y-intercept in our equation. In the slope-intercept form \( y = mx + b \), 'b' represents the y-intercept. So here, 'b' is 60. This can be interpreted as the starting point for our model. It tells us that in 1995, which is our base year (or when 'x' equals zero), 60% of U.S. adults were reading newspapers.
It's important to accurately identify the y-intercept because it sets the baseline from which we track subsequent changes over time. In many real-life cases, the y-intercept has a meaningful interpretation; in this case, it marks the initial state of newspaper readership before the start of the decline.
The initial readership percentage in our exercise, 60%, is the y-intercept in our equation. In the slope-intercept form \( y = mx + b \), 'b' represents the y-intercept. So here, 'b' is 60. This can be interpreted as the starting point for our model. It tells us that in 1995, which is our base year (or when 'x' equals zero), 60% of U.S. adults were reading newspapers.
It's important to accurately identify the y-intercept because it sets the baseline from which we track subsequent changes over time. In many real-life cases, the y-intercept has a meaningful interpretation; in this case, it marks the initial state of newspaper readership before the start of the decline.
Making Linear Model Predictions
One of the practical applications of a linear equation is to use it for linear model predictions. The ability to forecast future values based on historical trends is a powerful tool in many fields, including economics, biology, and sociology.
In our exercise, we use the equation \( y = -0.7x + 60 \) to predict future percentages of U.S. adults reading newspapers. By plugging in the number of years since 1995 into 'x', we can predict the corresponding 'y' value, which is the expected readership percentage.
For instance, to predict the readership in the year 2005, we would input '10' for 'x', since 2005 is 10 years after 1995. Doing so would result in a predicted percentage of 53%. This prediction is possible because the linear model assumes that the rate of change (the decrease) will continue consistently over time. However, it's important to keep in mind that linear models may not always hold true indefinitely, as rates can change due to unforeseen factors.
In our exercise, we use the equation \( y = -0.7x + 60 \) to predict future percentages of U.S. adults reading newspapers. By plugging in the number of years since 1995 into 'x', we can predict the corresponding 'y' value, which is the expected readership percentage.
For instance, to predict the readership in the year 2005, we would input '10' for 'x', since 2005 is 10 years after 1995. Doing so would result in a predicted percentage of 53%. This prediction is possible because the linear model assumes that the rate of change (the decrease) will continue consistently over time. However, it's important to keep in mind that linear models may not always hold true indefinitely, as rates can change due to unforeseen factors.
Other exercises in this chapter
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