Problem 54 - Intermediate Algebra

Question

Solve each formula for the specified variable. See Example 5. $$ P=2(w+h+l) \quad \text { for } h $$

Step-by-Step Solution

Answer

\( h = \frac{P}{2} - w - l \)

Step 1: Understand the Formula

The given formula is for the perimeter, denoted by \(P\), of a box where \(w\), \(h\), and \(l\) represent width, height, and length, respectively: \[ P = 2(w+h+l). \] Our goal is to solve for \(h\), which means we need to express \(h\) in terms of \(P\), \(w\), and \(l\).

Step 2: Isolate the Parentheses

First, divide both sides of the equation by 2 to isolate the term that contains \(h\):\[ \frac{P}{2} = w + h + l. \] This step removes the multiplication by 2 from the expression involving \(h\).

Step 3: Solve for \(h\)

Next, we need to isolate \(h\) by subtracting \(w\) and \(l\) from both sides of the equation:\[ \frac{P}{2} - w - l = h. \] Now \(h\) is expressed in terms of \(P\), \(w\), and \(l\).

Key Concepts

Perimeter Formula

The perimeter formula is often used in geometry to find the total distance around the outside of a given shape. In this context, the term "perimeter" refers to the sum of all sides encompassing the shape. This specific formula \[ P = 2(w + h + l) \] is designed to find the perimeter of a rectangular box's combined dimensions. Here:

• \( P \) represents the perimeter.
• \( w \), \( h \), and \( l \) correspond to the width, height, and length of the box, respectively.

The equation accounts for the fact that in a three-dimensional box, width, height, and length corresponding sides of the box, are considered twice. For a common understanding, when solving any geometric problem, identifying the correct formula applicable is crucial. This sets the foundation to correctly solve for the necessary variables later. Familiarizing oneself with different shapes and their perimeter calculations strengthens mathematical confidence.

Variable Isolation

Variable isolation is a fundamental technique in algebra where the objective is to solve for one variable by expressing it in terms of other variables. In a given equation, locating the specific variable to isolate becomes pivotal for a correct solution. For the equation \[ P = 2(w + h + l), \] we needed to isolate the variable \( h \). Here's how it was approached:

• First, dividing both sides of the equation by 2 gave us:\[ \frac{P}{2} = w + h + l. \] This operation moved us a step closer by simplifying the process.
• Next, the equation was further simplified by subtracting \( w \) and \( l \) from both sides, achieving:\[ h = \frac{P}{2} - w - l . \]This successfully isolated \( h \).

Isolating a variable helps in examining how the variable interacts with others within the equation. This is not only vital for solving equations but also enhances logical thinking, vital for tackling complex algebraic problems in the future.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations to simplify or extract particular information. It often involves a series of arithmetic operations such as addition, subtraction, multiplication, or division, applied systematically. In our scenario with the equation:\[ P = 2(w + h + l), \] we used algebraic manipulation in several steps:

• First, dividing by 2, was a strategic choice to handle the constant multiplier affecting \( h \).
• Next, the subtraction of \( w \) and \( l \) from both sides simplified the expression, making \( h \) the subject of the equation.

These actions are often referred to as 'manipulation' because they adapt the equation's form to serve a new purpose, such as solving for a specific variable. Understanding these tactics is critical for mastering algebra as these methods provide flexibility in exploring different problems. Practice is key, as familiarity with manipulation techniques can significantly enhance one's mathematical solutions repertoire.