Problem 64 - Prealgebra

Question

A rectangle has a perimeter of 11 inches. If the width is 2.5 inches, find the length.

Step-by-Step Solution

Answer

The length of the rectangle is 3 inches.

Step 1: Understanding the Problem

The problem involves a rectangle with a known perimeter of 11 inches and a known width of 2.5 inches. We need to find the length of this rectangle.

Step 2: Knowing the Formula for Perimeter

The formula for the perimeter of a rectangle is given by \[ P = 2(l + w) \] where \( P \) is the perimeter, \( l \) is the length, and \( w \) is the width.

Step 3: Substitute Known Values into the Formula

We know that the perimeter \( P = 11 \) inches and the width \( w = 2.5 \) inches. Substituting these values into the perimeter formula gives:\[ 11 = 2(l + 2.5) \]

Step 4: Solve for the Length

We need to solve the equation \( 11 = 2(l + 2.5) \). First, divide both sides by 2:\[ \frac{11}{2} = l + 2.5 \] Simplify the equation:\[ 5.5 = l + 2.5 \] Subtract 2.5 from both sides to isolate \( l \):\[ l = 5.5 - 2.5 \] \[ l = 3 \]

Key Concepts

Geometry

When we talk about geometry, we're discussing the branch of mathematics concerning shapes, sizes, and the properties of space. In this exercise, we are working with the shape known as a rectangle. Rectangles are defined by having opposite sides that are equal in length, and each angle is a right angle, which is 90 degrees. The perimeter of a geometric shape, like a rectangle, is the total length around the shape.
To find a rectangle's perimeter, we use the formula \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width. This formula essentially adds the length and width together, then multiplies by 2 because a rectangle has two lengths and two widths. Understanding this concept is crucial not only for solving rectangle-related problems but also for tackling various other shapes in geometry.

Solving Equations

The ability to solve equations is a core skill in math, often used to find unknown values based on given information. In this exercise, we use an equation derived from the perimeter formula to determine the unknown length of the rectangle.
Starting with the equation \( 11 = 2(l + 2.5) \), we aim to isolate the variable \( l \). The equation gives us a relationship between known and unknown values.

• First, we simplify by dividing both sides by 2, resulting in \( 5.5 = l + 2.5 \).
• Next, we solve for \( l \) by subtracting 2.5 from both sides, ending with \( l = 3 \).

Each step ensures both sides of the equation stay balanced, steadily bringing us closer to the solution. This process exemplifies a systematic approach to solving linear equations, highlighting methods like simplification and rearrangement.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In this exercise, we exercised algebraic methods to solve for an unknown variable, \( l \). This involves a few key elements: variables, constants, and operations.
In the equation \( 11 = 2(l + 2.5) \), "11" is a constant (a known value), "2.5" is another constant, and "\( l \)" is a variable (something we need to find).
Algebra involves manipulating these symbols to isolate and solve for the variable in question. By systematically transforming the equation, we found the length of the rectangle.
Understanding basic algebraic operations is fundamental for progressing in math as it allows us to handle more complex problems in geometry and beyond, ultimately enabling broader applications, like graphing and interpreting scientific data.