Problem 67 - Algebra A Combined Function

Question

A square has a side of length \(\frac{5}{x-2}\) meters. Express its perimeter as a rational expression.

Step-by-Step Solution

Answer

The perimeter is \( \frac{20}{x-2} \) meters.

Step 1: Identify the Formula for Perimeter of a Square

The perimeter of a square is calculated by multiplying the length of one side by 4, because a square has four equal sides. The formula is given by: \[ P = 4 imes ext{side length} \] where \( P \) is the perimeter of the square.

Step 2: Substitute the Given Side Length

Given the side length is \( \frac{5}{x-2} \) meters, substitute this expression into the perimeter formula. This gives: \[ P = 4 imes \left(\frac{5}{x-2}\right) \]

Step 3: Simplify the Expression

Multiply the 4 by the numerator of the expression for the side length. This results in: \[ P = \frac{4 imes 5}{x-2} \]

Step 4: Calculate the Final Perimeter

Carry out the multiplication in the numerator: \[ P = \frac{20}{x-2} \] Thus, the perimeter of the square, expressed as a rational expression, is \( \frac{20}{x-2} \) meters.

Key Concepts

Perimeter of a Square

The perimeter of a square refers to the total distance around the square. Since a square has four sides of equal length, calculating the perimeter is straightforward. You only need to multiply the length of one side by four. This is important because it allows us to quickly determine the perimeter once we know the side length. This is especially handy in algebraic problems involving rational expressions, as we often deal with the side length in terms of variables or fractions, like in our example.

Side Length

Understanding the side length is crucial, especially when we're dealing with rational expressions. The side length of a square is simply the measurement of one of its sides. In our case, the side length is given as a fraction: \( \frac{5}{x-2} \). This can be a bit tricky for some, as fractions introduce additional layers of complexity, such as the potential for undefined values. Here, the expression is undefined if \( x = 2 \), because that would make the denominator zero, which is not allowed in mathematics.

Simplifying Expressions

When simplifying expressions, the goal is to make them as straightforward as possible. Here, we start with the expression for the perimeter, which comes from multiplying the given side length \( \frac{5}{x-2} \) by 4. Simplifying involves applying the multiplication to just the numerator: \( 4 \cdot 5 \). This gives us \( \frac{20}{x-2} \). Each step must maintain the equivalence of the expression, ensuring that it accurately represents the same quantity in a more usable or simplified form. Keep in mind the importance of watching for any restrictions on the variable, like avoiding values that make the denominator zero.