Problem 13 - College Algebra with Modeling and Visualization

Question

Some workers use technology such as e-mail, computers, and multiple phone lines to work at home, rather than in the office. However, because of the need for teamwork and collaboration in the workplace, fewer employees are telecommuting than expected. The table lists telecommuters \(T\) in millions during year \(x\). $$\begin{array}{cccccc} x & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline T & 9.2 & 9.6 & 10.0 & 10.4 & 10.6 \end{array}$$ $$\begin{array}{cccccc} \hline x & 2002 & 2003 & 2004 & 2005 & 2006 \\ T & 11.0 & 11.1 & 11.2 & 11.3 & 11.4\end{array}$$ Find a function \(f\) that models the data, where \(x=1\) corresponds to \(1997, x=2\) to \(1998,\) and so on.

Step-by-Step Solution

Answer

The function that models the data is \(f(x) = 0.2444x + 8.9556\).

Step 1: Understanding the Problem

We need to find a function that models the relationship between year \(x\) and the number of telecommuters \(T\). Here, \(x\) starts from 1 in 1997 and increases by one each year, with corresponding values of \(T\) for each year.

Step 2: Choosing the Type of Function

The data for telecommuters \(T\) increases gradually each year. A linear function \(f(x) = mx + b\) is suitable for modeling data that shows a steady increase or decrease over time. Hence, a linear function is chosen to model \(T\) as a function of \(x\).

Step 3: Calculating the Slope

To find the slope \(m\), we use the slope formula \((y_2 - y_1)/(x_2 - x_1)\). Select two example points, for instance, \((1, 9.2)\) and \((10, 11.4)\):\[ m = \frac{11.4 - 9.2}{10 - 1} = \frac{2.2}{9} \approx 0.2444 \]

Step 4: Finding the Intercept

Use one point, such as \((1, 9.2)\), to solve for the intercept \(b\) in the equation \(f(x) = mx + b\):\[ 9.2 = 0.2444 \times 1 + b \]\[ b = 9.2 - 0.2444 \approx 8.9556 \]

Step 5: Writing the Linear Function

Substitute \(m\) and \(b\) back into the equation \(f(x) = mx + b\):\[ f(x) = 0.2444x + 8.9556 \]

Step 6: Verifying the Function

Check that this function approximates other data points in the table. For example, for \(x = 5\), which corresponds to the year 2001:\[ f(5) = 0.2444 \times 5 + 8.9556 = 10.1776 \]This value is close to the actual value of 10.6, confirming that the model is reasonable.

Key Concepts

Slope Calculation

The slope is one of the most crucial aspects when dealing with linear functions. It's the rate of change that tells us how much the response variable changes with a unit change in the predictor. In our example, this is how many telecommuters are added each year. To calculate the slope, you pick two data points from your dataset. Consider the points

• (1, 9.2) for the year 1997
• (10, 11.4) for the year 2006

With these, use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the values, and you'll get:\[ m = \frac{11.4 - 9.2}{10 - 1} = \frac{2.2}{9} \approx 0.2444 \]This shows each year, approximately 0.2444 million more people telecommuted than the previous year. It gives us the growth trend of the telecommuter data.

Intercept Determination

The intercept in a linear equation is where the line crosses the y-axis. This value represents the starting point when the predictor is zero, though in our example, it corresponds to the transformed starting year.Given the equation of a line as:\[ f(x) = mx + b \]we substitute one of our data points and the calculated slope to solve for the intercept, \( b \). Let's use the point (1, 9.2): \[ 9.2 = 0.2444 \times 1 + b \]Solving for \( b \) gives:\[ b = 9.2 - 0.2444 = 8.9556 \]So, the intercept is 8.9556, suggesting that around 8.9556 million people would have telecommuted at the beginning of 1997 in the context of our linear model. This fixes the point from which our line starts its journey across the graph.

Data Analysis

Data analysis involves understanding trends and making predictions based on the dataset. Here, the aim is to model the change in telecommuting rates over years, which involves analyzing how the number of telecommuters varied annually. We look at the tabulated data spanning from 1997 to 2006 and observe the consistent rise in telecommuters each year. Such a visible, regular increase is suggestive of a linear relationship, which makes a linear function appropriate for modeling these points. Conducting an analysis of this growth helps in:

• Identifying patterns: The consistent increase safeguarded by our slope.
• Predicting future trends: Foreseeing how many telecommuters exist beyond 2006 if the trend continues.
• Validating the model: Comparing estimated and actual data to ensure our linear function provides close approximations.

Linear Equations

Linear equations are fundamental in modeling relationships between variables. In our context, a linear equation models the evolution of telecommuting over time using the formula: \[ f(x) = mx + b \] Having calculated the slope \( m \approx 0.2444 \) and the intercept \( b \approx 8.9556 \), the equation becomes: \[ f(x) = 0.2444x + 8.9556 \]This equation can predict the number of telecommuters for any given year. By substituting the year as its corresponding \( x \)-value (e.g., \( x = 5 \) for 2001), you can estimate the telecommuters:\[ f(5) = 0.2444 \cdot 5 + 8.9556 = 10.1776 \]Such equations are not only used in historical data analysis but also serve as powerful tools for forecasting and strategic planning.