Problem 30 - Intermediate Algebra

Question

The length of a rectangle is twice the width of the rectangle. Given that the perimeter of the rectangle is 72 centimeters, find the dimensions.

Step-by-Step Solution

Answer

Width is 12 cm, length is 24 cm.

Step 1: Understand the Problem Statement

You are given a rectangle where the length is twice its width and the total perimeter is 72 centimeters. Let the width of the rectangle be denoted as \( w \) centimeters.

Step 2: Express the Relationship

According to the problem, the length \( l \) of the rectangle is twice the width. So, we can write \( l = 2w \).

Step 3: Use the Perimeter Formula

The formula for the perimeter of a rectangle is given by \( P = 2(l + w) \). Since the perimeter is 72 cm, we can set up the equation: \( 2(l + w) = 72 \).

Step 4: Substitute the Expressed Relationship

Substitute \( l = 2w \) into the perimeter equation, giving us: \( 2(2w + w) = 72 \).

Step 5: Simplify the Equation

Simplify inside the parentheses: \( 2(3w) = 72 \), which simplifies further to \( 6w = 72 \).

Step 6: Solve for \( w \)

To solve for \( w \), divide both sides by 6: \( w = \frac{72}{6} = 12 \).

Step 7: Find \( l \) Using \( w \)

Using the equation \( l = 2w \) and substituting \( w = 12 \), we find \( l = 2 \cdot 12 = 24 \).

Step 8: Write the Solution

The width of the rectangle is 12 centimeters, and the length is 24 centimeters.

Key Concepts

Understanding Rectangles

A rectangle is a four-sided polygon characterized by its opposite sides being equal in length. It is a type of quadrilateral, specifically a parallelogram, where each interior angle is 90 degrees. Rectangles are everywhere in real life—from books to screens, to doors—and understanding their properties is vital.

Key features of a rectangle include:

• Two pairs of parallel sides
• Opposite sides are equal in length
• Four right angles (each measuring 90 degrees)

These characteristics make rectangles very manageable for calculations, especially when working with their dimensions.

Exploring the Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the outside, and it's an essential aspect to know when determining the size of a rectangle. Calculating a rectangle's perimeter involves understanding and applying its formula effectively.

The formula for the perimeter (P) is:

• \[ P = 2(l + w) \]
• Where \( l \) is the length and \( w \) is the width

This formula highlights that a rectangle is so balanced that the entire measurement essentially folds down to the total of just two pairs of lengths, multiplied by two, because there are equal sides.

In practical terms, if we know any two dimensions (length and width), calculating the perimeter is a simple exercise, as was shown in the solution provided for the rectangle with a perimeter of 72 centimeters.

Solving Equations in Algebra

Solving equations is a cornerstone of intermediate algebra, a mathematical skill used to find unknown values. In the case of rectangles, we often have to solve equations to find missing dimensions.

Let's break down how the original problem was approached:

• Understanding relationships between dimensions, where one length was double the width (\( l = 2w \))
• Using a known value (perimeter = 72 cm) to form an equation, \( 2(2w + w) = 72 \)
• Simplifying and solving this equation guided us to the width \( w \)

This process involves both substituting known information into formulas and simplifying them to solve for the unknowns—skills that are vital not only in math, but across disciplines. Understanding concepts like solving equations helps in visualizing and tackling various mathematical and real-world problems with confidence.