Problem 12 - Elementary Algebra

Question

For Problems 11-32, use the geometric formulas given in this section to help solve the problems. (Objective 3 ) If the perimeter of a rectangle is 80 centimeters and its length is 24 centimeters, find its width.

Step-by-Step Solution

Answer

The width of the rectangle is 16 centimeters.

Step 1: Understand the formula

The formula for the perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width of the rectangle.

Step 2: Plug in known values

We know the perimeter \( P = 80 \) centimeters and the length \( l = 24 \) centimeters. Substitute these values into the formula: \( 80 = 2(24 + w) \).

Step 3: Simplify the equation

To simplify, divide both sides of the equation by 2: \( 40 = 24 + w \).

Step 4: Solve for the width

Subtract 24 from both sides to isolate the width \( w \): \( 40 - 24 = w \). This gives \( w = 16 \) centimeters.

Key Concepts

Rectangle Geometry

A rectangle is a simple yet fundamental shape in geometry, characterized by having four right angles and opposite sides equal in length. Rectangle geometry is essential for understanding many geometric calculations. Let’s break down the important parts of a rectangle’s geometry:

• Sides: A rectangle has two pairs of opposite sides. The longer side is usually referred to as the length (\( l \)), and the shorter one as the width (\( w \)).
• Angles: All internal angles in a rectangle are 90 degrees, which makes it a perfect example of a quadrilateral.
• Diagonals: In a rectangle, diagonals are equal in length and bisect each other. This characteristic helps in further mathematical calculations.
• Perimeter: The perimeter is the total distance around the rectangle and is calculated by adding the lengths of all sides.

Understanding these basic features lays the groundwork for solving more complex geometrical problems. Exploring rectangle geometry helps in visualizing and solving problems involving dimensions and measurements.

Solving Equations

Solving equations is a critical skill that allows us to find unknown values in mathematical expressions. In this context, we are dealing with a simple linear equation involving the perimeter of a rectangle.
Here’s how we approached solving it:

• First, understand the equation we’re working with: \[P = 2(l + w)\]This equation relates the perimeter (\( P \)), length (\( l \)), and width (\( w \)).
• We inserted known values into the equation. These include the perimeter of 80 centimeters and the length of 24 centimeters.
• To isolate the unknown (\( w \)), we simplified the equation step by step. Divide both sides by 2 to make calculations easier. Then, subtract the known length from both sides to find the width.

Breaking down the equation into manageable steps allows us to systematically find the unknown value. Mastering this process is key to handling various algebra problems effectively.

Geometric Formulas

Geometric formulas are powerful tools that help us solve a wide range of problems concerning shapes and measurements. They allow us to derive unknown properties of geometric figures by using known values.
For rectangles, one important formula is for calculating the perimeter:\[P = 2(l + w)\]Here's how this formula assists in problem-solving:

• Simple substitution: We replace the variables with their known values. This step involves substituting the known perimeter and length to solve for width.
• Simplification: After substituting, simplify the equation by performing arithmetic operations. It's important to apply operations like division or subtraction strategically to isolate the unknown.
• Relevance: This perimeter formula can apply to any rectangle regardless of size, making it a versatile tool in geometry.

Understanding and applying geometric formulas not only solve specific problems but also provide a framework for tackling similar problems with different figures. Mastery of these formulas broadens one's mathematical toolkit significantly.