Problem 95 - Beginning and Intermediate Algebra

Question

The perimeter of a figure is the distance around the figure. The expression \(2 l+2 w\) represents the perimeter of a rectangle when \(l\) is its length and \(w\) is its width. Find the perimeter of the following rectangle by substituting 8 for \(l\) and 6 for \(w\).

Step-by-Step Solution

Answer

The perimeter of the rectangle is 28.

Step 1: Identify the Formula

The formula to find the perimeter of a rectangle is given by the expression \(2l + 2w\), where \(l\) is the length and \(w\) is the width.

Step 2: Substitute Values

Substitute the given values into the formula: \(l = 8\) and \(w = 6\). This makes the expression \(2(8) + 2(6)\).

Step 3: Calculate Each Term

Calculate the multiplication for each term of the expression: \(2(8)\) equals 16, and \(2(6)\) equals 12.

Step 4: Add the Results

Add the results from Step 3: 16 and 12, which gives you the total perimeter, \(16 + 12 = 28\).

Key Concepts

Rectangle Formula

The concept of the rectangle formula is essential when calculating the perimeter of a rectangle. Perimeter is the total distance around a shape. For a rectangle, this involves adding up the lengths of all four sides. A rectangle has two sets of sides: two lengths (often noted as \( l \)) and two widths (\( w \)). Thus, the formula to find the perimeter is expressed as:\[ 2l + 2w \]This formula sums up the lengths of two long sides (\( l \) each) and two short sides (\( w \) each). It's a clear and straightforward method for computing the complete boundary of a rectangle, simply by knowing its length and width. Remember, understanding this formula is key to solving perimeter problems efficiently.

Substituting Values

Substituting values into a formula helps transform a general expression into a specific numerical one. This step is crucial as it tailors the formula to the problem at hand. In our rectangle perimeter formula, \( 2l + 2w \), we substitute the given numbers for \( l \) and \( w \). For instance:

• Length, \( l = 8 \)
• Width, \( w = 6 \)

This changes our formula to:\[ 2(8) + 2(6) \]Substitution thus personalizes mathematical expressions to be applied to real-world problems. Always double-check you're using the correct numbers for each variable, as accuracy here ensures a correct perimeter calculation.

Mathematical Expression

A mathematical expression involves various operations to represent numbers and variables compactly. In this case, the expression \( 2l + 2w \) includes multiplication and addition. Breaking down the expression helps clarify how each part contributes to the outcome.

Understand the Operations

Firstly, \( 2l \) means multiplying the length \( l \) by 2. Similarly, \( 2w \) means multiplying the width \( w \) by 2. These operations ensure each side of the rectangle is counted twice, as there are two lengths and two widths in a rectangle.Expressions like this are the foundation of algebra, allowing complex ideas to be simplified into manageable parts.

Arithmetic Operations

Arithmetic operations are basic mathematical calculations including addition, subtraction, multiplication, and division. In calculating the perimeter, we primarily focus on multiplication and addition.

Break Down Operations

• **Multiplication:** Calculate \( 2 \times 8 \) to get 16, and \( 2 \times 6 \) to get 12.
• **Addition:** Add these products together, \( 16 + 12 = 28 \).

Each step directly follows from substituting values into the formula. Performing these operations accurately ensures you derive the correct perimeter of the rectangle. It's always helpful to double-check calculations to avoid errors, keeping your arithmetic precise and reliable.