Problem 154 - College Algebra Essentials

Question

A piece of wire is 8 inches long. The wire is cut into two pieces and then each piece is bent into a square. Find the length of each piece if the sum of the areas of these squares is to be 2 square inches.

Step-by-Step Solution

Answer

The lengths of the two pieces are \(x\) and \(y\) inches, with \(x\) and \(y\) being the solutions of the system of equations. Remember to verify that \(x\) and \(y\) are positive real numbers.

Step 1: Define the variables

Let's denote the two pieces of wire using variables \(x\) and \(y\). Thus, we can represent the total length of the wire as \(x + y = 8 \).

Step 2: Formulate the equation for the sum of the areas of the squares

The area of a square made with a piece of wire of length \(x\) is \((x/4)^2\) and likewise, the area of a square made with a piece of wire of length \(y\) is \((y/4)^2\). Hence, the sum of the areas of the squares is \((x/4)^2 + (y/4)^2 = 2\).

Step 3: Solve the system of equations

Now we have a system of two equations. We can substitute \(y = 8 - x\) from Step 1 to the equation in Step 2 and solve for \(x\).

Step 4: Find the length of each piece

After finding \(x\), substitute the value of \(x\) into \(y = 8 - x\) to find the second piece's length \(y\). If the resulting \(x\) and \(y\) are positive real numbers, then they are the lengths of the two pieces of wire.

Key Concepts

Quadratic Equations

Quadratic equations are at the heart of many algebraic problems, and they appear frequently in various applications, including geometry. A quadratic equation can be recognized by its standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable or unknown quantity. The solutions to a quadratic equation, also known as the roots, can be found using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), or by factoring and completing the square.

In the context of our exercise, we encounter a quadratic equation when we attempt to ascertain the lengths of two pieces of a wire, based on the areas of squares formed from them. Understanding how to manipulate and solve quadratic equations is crucial in finding the correct length of each piece of wire to meet the conditions given in the problem.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They allow us to represent relationships between quantities abstractly. In algebra, we commonly use expressions to define equations that model real-life situations.

For instance, in our exercise, the total length of wire is initially given as 8 inches. When we cut the wire into two pieces and represent each piece with variables \(x\) and \(y\), we create an algebraic expression \(x + y\) to symbolize the total length. The beauty of algebraic expressions is their versatility; they can vary to model different scenarios, which, in our case, helps us establish the algebraic relationship needed to solve for both \(x\) and \(y\) based on the areas of the squares they form.

Area of Squares

The area of a square, which is the amount of space enclosed within its boundaries, is a fundamental concept in geometry. It is calculated by squaring the length of one side of the square. The algebraic expression to represent the area of a square with side length \(s\) is \(A = s^2\).

In our exercise, we are dealing with squares formed by bending pieces of wire. If a piece of wire is cut and then bent into a square, the perimeter of that square is equal to the length of the wire used. Therefore, if a piece of wire is \(x\) inches long, each side of the square will be \(x/4\) inches, resulting in an area of \(\left(\frac{x}{4}\right)^2\). This concept enables us to relate the areas of the squares to the lengths of the wire pieces and is key to setting up the equation needed to solve our problem.