Problem 69 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(P=2 l+2 w\) for \(w\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Formula

The formula given is for the perimeter of a rectangle, where \(P\) is the perimeter, \(l\) is the length, and \(w\) is the width. We need to manipulate this formula to solve for \(w\).

Step 2: Isolate Terms with the Variable w

Begin by isolating terms that contain the variable \(w\) on one side. Starting with the equation \(P = 2l + 2w\), subtract \(2l\) from both sides to get: \[P - 2l = 2w\]

Step 3: Solve for w by Division

Now that the term with \(w\) is isolated, divide both sides of the equation by 2 to solve for \(w\): \[w = \frac{P - 2l}{2}\]

Step 4: Conclude the Solution

The variable \(w\) is isolated, and we have the formula expressed explicitly in terms of \(P\) and \(l\): \[w = \frac{P - 2l}{2}\]

Key Concepts

Perimeter of a Rectangle

Understanding the perimeter of a rectangle is crucial when solving related formulas. The perimeter is the distance around the edge of a rectangle. To calculate it, you add the lengths of all four sides. For a rectangle, this is expressed by the formula: \[ P = 2l + 2w \]Where:- \( P \) represents the perimeter- \( l \) is the length- \( w \) is the widthEach pair of length \( l \) and width \( w \) are multiplied by 2 because there are two of each side in a rectangle. Knowing this formula allows you to find the perimeter if length and width are known or vice versa. It forms the basis for many geometric problems that require understanding the concept of perimeter.

Variables

Variables are symbols used to represent unknown values in equations and formulas. In the context of our rectangle perimeter formula \( P = 2l + 2w \), the variables used are \( P \), \( l \), and \( w \). These variables help describe dynamic aspects of a rectangle. - \( P \): can change with the shape’s size- \( l \): varies with the length of the rectangle- \( w \): represents the rectangle's widthUnderstanding variables helps in manipulating formulas and solving for unknown measures. They allow equations to be flexible and applicable to different situations, offering a systematic way to develop solutions.

Manipulating Formulas

Manipulating formulas involves changing the structure of an equation to solve for a specific variable. It requires applying basic algebraic operations like addition, subtraction, multiplication, and division. In our problem, to solve for \( w \) in the formula \( P = 2l + 2w \), follow these steps:- First, subtract \( 2l \) from both sides: \[ P - 2l = 2w \]- Next, divide everything by 2 to isolate \( w \): \[ w = \frac{P - 2l}{2} \]Each step carefully manipulates the equation, maintaining balance by performing the same operation on both sides. This technique is essential for isolating a specific variable, making it easier to find its value given other variables.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the building blocks of equations and formulas. In the rectangle's perimeter formula, \( P = 2l + 2w \), each part is an algebraic expression:- \( 2l \) and \( 2w \): each term represents a part of the perimeter- \( P - 2l = 2w \): resulting expression after manipulationBy understanding how these expressions work, you can rearrange terms to solve for different variables. Expressions like \( w = \frac{P - 2l}{2} \) exemplify how a complex equation can be transformed into a simple equation for easy interpretation and solution. Mastery of algebraic expressions is crucial for anyone learning algebra, as it facilitates breaking down and solving even the most complicated problems.