Problem 84 - College Algebra and Calculus: An Applied Approach

Question

Solve for the indicated variable. Perimeter of a Rectangle Solve for \(l\) in \(P=2 l+2 w\).

Step-by-Step Solution

Answer

To find the length \(l\), we can use the formula \(\frac{P - 2w}{2} = l\).

Step 1: Subtract from Both Sides

In order to isolate \(l\), we first subtract \(2w\) from both sides of the equation, which gives us, \(P - 2w = 2l\).

Step 2: Divide by the Coefficient of \(l\)

Then, to further isolate \(l\), we divide both sides of the equation by 2, the coefficient of \(l\). So, \(\frac{P - 2w}{2} = l\).

Key Concepts

Rectangle Perimeter

The perimeter of a rectangle is an important geometric concept used in various real-world applications. It refers to the total distance around the boundary of a rectangle. You can calculate the perimeter of a rectangle using the formula \( P = 2l + 2w \). Here, \( l \) stands for the length of the rectangle, and \( w \) stands for the width.
This formula essentially adds up the lengths of all four sides of the rectangle. Since a rectangle has opposite sides that are equal, the perimeter simplifies to twice the sum of its length and width.

• For instance, if a rectangle has a length of 5 units and a width of 3 units, its perimeter can be calculated as \( P = 2(5) + 2(3) = 16 \) units.

The understanding of how to compute the perimeter is crucial in problems involving dimensions of a rectangle, especially when manipulating formulas to solve for an unknown variable.

Isolation of Variables

Isolation of variables is a mathematical technique used to solve equations by expressing a particular variable in terms of the others. This technique helps in finding the value of an unknown variable by getting it alone on one side of the equation.
In our scenario, we want to isolate \( l \) from the equation \( P = 2l + 2w \).
Here's how it works:

• **Step 1:** We start by removing any terms that do not contain \( l \) from the side where \( l \) is located. For our equation, this involves subtracting \( 2w \) from both sides, giving us \( P - 2w = 2l \).
• **Step 2:** We then eliminate the coefficient of \( l \) by dividing all terms by 2, resulting in \( l = \frac{P - 2w}{2} \).

This sequence of operations ensures that \( l \) is isolated and can be directly solved once the other parameters \( P \) and \( w \) are known.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and arithmetic operations that together define a particular value or relationship.
An expression doesn't have an equality sign like an equation, but can be manipulated to solve equations.
In the context of the rectangle perimeter problem, the expression \( 2l + 2w \) represents the perimeter in terms of length and width.

• **Understanding Expressions:** Grasping such expressions involves knowing what each symbol stands for and how they work together in the formula.
• **Manipulating Expressions:** When solving for a variable, the goal is to rewrite the expression so that the desired variable is by itself on one side of the equation.

Expressions can be simplified, expanded, or rearranged based on the operations applied to achieve the isolation of a particular variable. This understanding is pivotal in effectively solving equations and real-world mathematical problems.