Problem 88 - College Algebra and Calculus: An Applied Approach

Question

From 1994 to 2005, the total energy imports \(y\) (in quadrillions of Btu's) to the United States increased in a pattern that was approximately linear (see figure). Find the least squares regression line \(y=a t+b\) for the data shown in the figure by solving the following system using matrices. Let \(t\) represent the year, with \(t=0\) corresponding to 1994 . \(\left\\{\begin{array}{l}12 b+66 a=334.80 \\ 66 b+506 a=1999.91\end{array}\right.\) Use the result to predict the total energy imports in \(2010 .\) Is the estimate reasonable? Explain. (Source: Energy

Step-by-Step Solution

Answer

Through solving the system of linear equations using matrices, the coefficients of the least squares regression line can be obtained for future predictions. After calculations, the estimate value for the total energy imports in 2010 can be found. The reasonability of this estimate should be evaluated considering both the mathematical pattern and real-world context.

Step 1: Formulate the matrices

The system of linear equations provided in the exercise can be presented as a matrix equation \(Ax = b\), where \(A\) is the coefficient matrix, \(x\) is the variable matrix, and \(b\) is the ordinates matrix. So, we have: \[A=\left[\begin{matrix}12 & 66 \ 66 & 506\end{matrix}\right]\], \[x=\left[\begin{matrix}b \ a\end{matrix}\right]\], \[b=\left[\begin{matrix}334.8 \ 1999.91\end{matrix}\right]\]

Step 2: Compute the inverse of the coefficient matrix

The inverse of matrix \(A\) is given by the formula \[A^{-1}=\frac{1}{ad-bc}\left[\begin{matrix}d & -b \ -c & a\end{matrix}\right]\] where \[A=\left[\begin{matrix}a & b \ c & d\end{matrix}\right]\]. Thus, the inverse matrix \(A^{-1}\) is given by \[A^{-1}=\frac{1}{(12 * 506) - (66 * 66)}\left[\begin{matrix}506 & -66 \ -66 & 12\end{matrix}\right]\]

Step 3: Multiply the inverse matrix with the ordinates matrix

We can find matrix \(x\) by multiplying the inverse of matrix \(A\) and matrix \(b\). This gives us the values of \(b\) and \(a\), which represent the intercept and slope of the least squares regression line respectively. The calculation is as follows: \[x=A^{-1}b\]

Step 4: Predict the total energy imports in 2010

The least squares regression line can be used to predict the total energy imports in 2010. The equation of the line is given by \(y=a*t+b\), where \(t\) represents the year (with \(t=0\) corresponding to 1994). Thus, for 2010 (\(t=16\)), the total energy imports \(y\) can be calculated by substituting the values of \(a\), \(b\) and \(t\) into the equation.

Step 5: Evaluate the reasonability of the estimate

The reasonability of the estimate can be evaluated based on the concept of the least squares regression line. Since the line corresponds to the best linear fit for the data from 1994 to 2005, the predicted value for 2010 would be reasonable if it follows the same linear pattern. The physical aspects (political, economic, technological) should be considered for a comprehensive judgement.

Key Concepts

Matrix Algebra

Matrix Algebra is a powerful tool in mathematics that simplifies complex system calculations. Here, we use matrices to solve a system of linear equations representing a real-world problem involving energy imports to the United States.
In our problem, the task is to find a regression line that fits the given data using matrices. We start by formulating our equations in matrix form, which makes calculations more structured and efficient.

• The system of equations can be represented as a matrix equation of the form \(Ax = b\).
• Matrix \(A\) is the coefficient matrix obtained from the constants in the equations, while \(x\) represents the unknowns (slope and intercept in the regression), and matrix \(b\) contains the constants on the right side of the equations.

This method of organizing equations helps in solving multiple equations simultaneously using matrix operations like multiplication and inversion.

Linear Equation

A Linear Equation is a vital concept in mathematics, expressing the relationship between two quantities. Linear equations are straight lines on a graph, defined by the equation \(y = ax + b\), where \(a\) is the slope, and \(b\) is the y-intercept.
In the context of our exercise, the linear equation represents the trend of total energy imports over time. Our goal was to find the least squares regression line, which mathematically minimizes the error (or the distance) between the data points and the line.

• We used a system of linear equations, derived from the sum of the squares of the vertical distances of the different data points from the line, to determine the values of \(a\) and \(b\).
• Subsequently, by solving this system using matrix algebra, we can determine the intercept (\(b\)) and the slope (\(a\)) of the best-fit line that models the given data.

This equation helps to summarize and generalize the observed data into an easily interpretable form.

Data Prediction

Data Prediction involves using mathematical models to forecast future data points based on historical data. It is an essential aspect of utilizing mathematical equations effectively to anticipate trends and make informed decisions.
In this case, once the least squares regression line was derived, it was used to predict energy imports for the year 2010.

• By inserting \(t = 16\) (since 1994 represents \(t = 0\) and 2010 represents \(t = 16\)), into the equation \(y = at + b\), we obtain a projected value for the total energy imports in 2010.
• This prediction helps to give insight into how trends continue or change based on past data.

The accuracy of such a prediction depends on the assumption that the established linear trend will remain consistent over time.

Energy Imports

Energy Imports refer to the total amount of energy brought into a country from external sources, measured in quadrillions of Btu's in this case. The pattern of energy imports can reveal significant insights about a country's energy needs and policies.
Between 1994 and 2005, U.S. energy imports showed an approximately linear increase, signifying consistent growth.

• This trend might result from numerous factors including economic growth, increased energy consumption, or changes in domestic production.
• Understanding this trend helps in creating policies to manage energy resources efficiently and plan for future energy supply needs.

The study of such trends, as highlighted by solving the regression problem, is crucial for energy economists and policy-makers to forecast future demands and address potential import dependencies.