Problem 85 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. \(P=2 l+2 w ; \quad\) for \(w\)

Step-by-Step Solution

Answer

The formula for width \( w \) is \( w = \frac{P - 2l}{2} \).

Step 1: Understand the Problem

The equation given is for the perimeter \( P \) of a rectangle, where \( l \) is the length and \( w \) is the width. We are required to solve for \( w \).

Step 2: Isolate Terms Involving w

Start by moving terms involving \( l \) to the other side of the equation. Subtract \( 2l \) from both sides: \[ P - 2l = 2w \] This isolates the term \( 2w \) on one side of the equation.

Step 3: Solve for w

Now, divide both sides of the equation by 2 to solve for \( w \): \[ w = \frac{P - 2l}{2} \] This gives the formula for \( w \) in terms of \( P \) and \( l \).

Key Concepts

Understanding Perimeter

Perimeter is a term often used in geometry to indicate the total distance around the edge of a two-dimensional shape. For any shape, calculating the perimeter involves adding up the lengths of all its sides.
For a rectangle, the formula for the perimeter, denoted as \( P \), is given by:

• \( P = 2l + 2w \)

where \( l \) represents the length, and \( w \) is the width of the rectangle. You might wonder why we multiply by two. It's simple: a rectangle has two lengths and two widths. Adding them together gives us the total perimeter.
Understanding this concept is essential, as it applies to various real-world situations, like fencing a yard or framing a picture. You use the perimeter to know how much material you will need for the boundary of any rectangular space.

Properties of a Rectangle

A rectangle is a four-sided polygon known for its right angles. Each of its interior angles is 90 degrees. This feature makes it one of the most straightforward shapes to work with in geometry. The opposite sides of a rectangle are equal and parallel, which simplifies calculations related to its dimensions significantly.
Here's a quick rundown of properties specific to rectangles.

• Two pairs of equal sides: the length (\( l \)) and the width (\( w \)).
• Two pairs of parallel sides: enhances stability and balance in design.
• Diagonals that are equal in length: this can be used in various advanced applications like dividing the shape into smaller, congruent triangles.

These properties form the core foundation when calculating measurements like area and perimeter, providing a roadmap for isolating variables and understanding the geometric shape in depth.

Isolation of Variables Explained

Isolation of variables is a key technique in algebra that allows us to solve equations for one specific variable. In the context of the given exercise, you want to solve for the variable \( w \) (width) in the equation for the perimeter of a rectangle.
The process starts by ensuring that all terms involving your variable of interest are on one side of the equation. This is achieved by eliminating other terms from that side which don't contain the variable. For example:

• Start with the equation: \( P = 2l + 2w \)
• Subtract \( 2l \) from both sides to get: \( P - 2l = 2w \)

Now, the term \( 2w \) is isolated on one side, making the next step straightforward.
The final step involves dividing each term by the coefficient of \( w \) (which is 2 in this case), giving the formula: \( w = \frac{P - 2l}{2} \). This isolated formula can be readily used for any further calculation or substitution involving \( w \).
Mastering isolation of variables not only aids in solving geometric problems but is also fundamental in various fields of science, engineering, and economics, where mathematical modeling is essential.