Problem 260 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. Find the width of a rectangle with perimeter 92 and length \(19 .\)

Step-by-Step Solution

Answer

The width of the rectangle is 27.

Step 1: Understand the Perimeter Formula for a Rectangle

The formula for the perimeter of a rectangle is given by: \[ P = 2L + 2W \]where \( P \) is the perimeter, \( L \) is the length, and \( W \) is the width.

Step 2: Substitute the Known Values

We are given the perimeter \( P = 92 \) and the length \( L = 19 \). Substitute these values into the perimeter formula: \[ 92 = 2(19) + 2W \]

Step 3: Simplify the Equation

Multiply the length by 2 and rewrite the equation:\[ 92 = 38 + 2W \]

Step 4: Solve for \( W \)

Subtract 38 from both sides to isolate the term with \( W \):\[ 92 - 38 = 2W \]which simplifies to:\[ 54 = 2W \]Next, divide both sides by 2 to solve for \( W \):\[ W = \frac{54}{2} \]\[ W = 27 \]

Key Concepts

Perimeter Formula

The concept of the perimeter is fundamental in geometry. The perimeter of a shape is the total length of its boundary. For rectangles, the perimeter formula is specific and can be written as:\[ P = 2L + 2W \]In this formula, \(P\) stands for Perimeter, \(L\) represents the Length, and \(W\) represents the Width of the rectangle. This formula works because a rectangle has two pairs of equal opposite sides.To break it down:

• Each rectangle has two lengths and two widths.
• Adding both lengths together gives \(2L\).
• Adding both widths together results in \(2W\).

By summing these combined lengths and widths, we obtain the total perimeter of the rectangle.

Solving Equations

Solving equations is a critical skill in algebra. When you are given an equation, the goal is to find the value of the unknown variable that makes the equation true. Let's look at the equation from our exercise:\[ 92 = 2(19) + 2W \]Here, the perimeter (92) is already known, and the length (19) is also given. We need to find the width (W). Steps to solve the equation:

• First, substitute the given values into the perimeter formula.
• Simplify the equation by performing the arithmetic operations.
• Isolate the variable we are solving for, in this case, \(W\).

This transforms the problem into a straightforward algebraic equation, helping us focus solely on solving for \(W\).

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to solve for an unknown variable. In our exercise, we start with the equation:\[ 92 = 38 + 2W \]To isolate \(W\), we need to perform actions that simplify the equation:1. Subtract 38 from both sides to eliminate the constant on the right.\[ 92 - 38 = 2W \]This results in:\[ 54 = 2W \]2. Next, divide both sides by 2 to solve for \(W\):\[ W = \frac{54}{2} \]This simplifies to:\[ W = 27 \]

By following these steps, algebraic manipulation ensures that all the operations are performed correctly and logically, leading us to the correct solution.

Geometry of Rectangles

Understanding the geometry of rectangles helps us comprehend why the perimeter formula works and how different properties of rectangles are derived. A rectangle is a four-sided shape where opposite sides are parallel and equal in length.Some key properties include:

• Opposite sides are equal: This means each pair of opposite sides can be represented as \(L\) and \(W\).
• Right angles: Each of the four angles in a rectangle is a right angle (90 degrees).
• Diagonals are congruent: The two diagonals of a rectangle are of equal length.

Because rectangles have these consistent properties, we can use formulas like the perimeter formula directly and effectively. Applying this knowledge to real-world problems, such as finding the width when given the perimeter and length, becomes straightforward and intuitive.