Problem 29 - Elementary Algebra

Question

The length of a rectangle is twice that of its width. If the area of the rectangle is 72 square inches, then find the length and width.

Step-by-Step Solution

Answer

The width is 6 inches, and the length is 12 inches.

Step 1: Express the relationships using variables

Let's denote the width of the rectangle by \( w \) and the length by \( l \). According to the problem, the length is twice the width, so we have the equation \( l = 2w \).

Step 2: Write the area equation

The area of a rectangle is given by the formula \( A = l \times w \). Since the area of the rectangle is 72 square inches, we can substitute to get \( l \times w = 72 \).

Step 3: Substitute the length in the area equation

Replace \( l \) with \( 2w \) in the area equation. This gives us \( 2w \times w = 72 \).

Step 4: Solve for the width

The equation \( 2w \times w = 72 \) simplifies to \( 2w^2 = 72 \). Divide both sides by 2 to get \( w^2 = 36 \). Take the square root of both sides to find \( w = 6 \).

Step 5: Calculate the length

Use the relationship \( l = 2w \). Since \( w = 6 \), substitute to find \( l = 2 \times 6 = 12 \).

Key Concepts

Rectangle Dimensions

A rectangle is a simple geometric shape characterized by its four right angles and opposite sides equal in length. For any rectangle, knowing two key dimensions—length and width—is essential to calculate its area or perimeter.

• Length: Often the longer side of the rectangle. We denote it as "l" in equations.
• Width: Typically the shorter side, and denoted as "w."

Rectangles are commonly found in everyday objects like books, screens, and tables. Understanding the basic properties of rectangles provides a foundation for solving more complex problems related to shapes.

Length and Width Relationship

In any rectangle problem, the relationship between length and width can simplify finding unknown values. In this exercise, you learn that:
- The length (\( l \)) is twice the width (\( w \)).
This relationship can be expressed as an equation: \( l = 2w \).This equation helps reduce variables, ensuring easier calculations when solving for either dimension. When given one dimension, you can directly calculate the other using this relationship. It is a powerful tool frequently used in problems where dimensions depend directly on one another. Such algebraic expressions unveil direct proportionalities or dependencies between different parts of a shape.

Algebraic Equation

Algebraic equations are mathematical sentences that use letters to represent unknown numbers. In this problem, we use equations to find the unknown length and width of a rectangle when given an area.
### Forming the Equations1. The given area is 72 square inches, so we use the formula for rectangle area: \( A = l \times w \). 2. Substitute the length's expression from the length and width relationship: \( 2w \times w = 72 \).
### Solving the EquationsSimplify the equation: \( 2w^2 = 72 \). Solve for \( w \) by: - Dividing each side by 2: \( w^2 = 36 \).- Solving for \( w \) by taking a square root: \( w = 6 \).Finally, substitute \( w \) back to find \( l = 2 \times 6 = 12 \).
Algebraic equations provide a logical pathway for solving complex problems step by step. Understanding how to set up and manipulate these equations is fundamental in mathematics, helping solve various practical problems.