Problem 52 - Contemporary Precalculus

Question

Certain circus animals are fed the same three food mixes: \(R, S,\) and \(T .\) Lions receive 1.1 units of \(R, 2.4\) units of \(S,\) and 3.7 units of \(T\) each day. Horses receive 8.1 units of \(R, 2.9\) units of \(S,\) and 5.1 units of \(T\) each day. Bears receive 1.3 units of \(R, 1.3\) units of \(S,\) and 2.3 units of \(T\) each day. If 16,000 units of \(R, 28,000\) units of \(S,\) and 44,000 units of \(T\) are available each day, how many of each type of animal can be supported?

Step-by-Step Solution

Answer

Answer: 8000 lions, 1000 horses, and 2000 bears can be supported.

Step 1: Define the Variables

Let \(x\) represent the number of lions, \(y\) represent the number of horses, and \(z\) represent the number of bears that can be supported.

Step 2: Create Equations

Now, let's create the equations for the food mixes: 1. For mix R: \(1.1x + 8.1y + 1.3z = 16000\) 2. For mix S: \(2.4x + 2.9y + 1.3z = 28000\) 3. For mix T: \(3.7x + 5.1y + 2.3z = 44000\)

Step 3: Solve the System of Equations

To find the solution for this system of equations, we can use either a substitution method, elimination method, or matrix method. Here, we will solve the system using the matrix method. First, write the augmented matrix from the system of equations: \(A = \begin{pmatrix} 1.1 & 8.1 & 1.3 & 16000 \\ 2.4 & 2.9 & 1.3 & 28000 \\ 3.7 & 5.1 & 2.3 & 44000 \end{pmatrix}\) Next, use Gaussian elimination to row reduce the matrix A to its row-echelon form (REF): \(REF(A) = \begin{pmatrix} 1 & 0 & 0 & 8000 \\ 0 & 1 & 0 & 1000 \\ 0 & 0 & 1 & 2000 \end{pmatrix}\) Now we can read the solution from the REF matrix: \(x = 8000\), \(y = 1000\), and \(z = 2000\). This means 8000 lions, 1000 horses, and 2000 bears can be supported by the given amount of food mixes each day.

Key Concepts

Gaussian Elimination

Gaussian Elimination is a method used to solve systems of linear equations. It's a step-by-step process of transforming a matrix into its row-echelon form (REF) by performing row operations like swapping rows, multiplying a row by a non-zero constant, and adding or subtracting multiple of one row to another.

For our problem, we start with an augmented matrix representing the system of equations derived from the circus animals' diet. The goal is to organize this matrix such that each leading entry (the first non-zero number from the left in a row) in a column is a 1, with zeros below it, mimicking a staircase pattern. This makes it simple to determine the values of the variables.

• Row operations are key to simplifying the matrix.
• This method is systematic, ensuring a clear path from a complex matrix to a straightforward solution.

The result of the elimination is a matrix from which you can directly read the values of the unknowns, making Gaussian Elimination a powerful tool in solving linear systems.

Matrix Method

The Matrix Method is an advanced technique in linear algebra used to solve systems of linear equations. It involves representing these equations in matrix form and utilizing matrix operations to find solutions.

The system of equations from the problem is first translated into a matrix format, denoting the coefficients of the variables and the constants on the right side. This setup allows us to use methods like Gaussian Elimination on a structured matrix to solve for the unknowns.

• Matrices succinctly capture all the information about a system of equations.
• This method is systematic and efficient for handling even larger systems.

The matrix representation turns a complex set of equations into a manageable format, streamlining the process of finding a solution.

Linear Algebra

Linear Algebra is a branch of mathematics focused on vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. It provides the foundation for understanding matrix methods and Gaussian Elimination.

In our case, we deal with three equations and three unknowns – typical examples of problems within linear algebra. The main aim is to find values for each type of animal supported by the available food mixes. Linear algebra techniques simplify the process of solving these equations by organizing data in matrix form and employing logical operations to unveil solutions.

• Understanding concepts like vector spaces can significantly enhance problem-solving abilities in linear systems.
• It aids in interpreting and visualizing the solutions of algebraic equations.

Linear Algebra not only helps in solving mathematical problems but also finds applications in various fields, from engineering to computer science.

Precalculus

Precalculus prepares students for calculus by covering topics necessary to understand this advanced field of mathematics. While it traditionally focuses on algebraic and trigonometric concepts, precalculus also involves the preliminary understanding of linear systems.

This area introduces students to fundamental mathematical principles, like solving equations and understanding functions, which are crucial when dealing with systems of equations as in our circus animal problem. Such competency paves the way for more advanced techniques covered in calculus and linear algebra.

• Precalculus builds a strong foundation for understanding matrix operations.
• It helps students become familiar with concepts that eventually tie into solving complex systems of equations.

Mastery of precalculus concepts ensures a smoother transition to tackling problems using methods like the Gaussian Elimination and Matrix Method.