Problem 259 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. Find the length of a rectangle with perimeter 124 and width 38 .

Step-by-Step Solution

Answer

The length of the rectangle is 24.

Step 1: Understand the Perimeter Formula for a Rectangle

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

Step 2: Plug in the Given Values

We are given the perimeter \( P = 124 \) and the width \( w = 38 \). Substitute these values into the perimeter formula: \[ 124 = 2(l + 38) \]

Step 3: Solve for the Length

First, solve for \( l + 38 \): \[ 124 = 2(l + 38) \] Divide both sides by 2: \[ 62 = l + 38 \] Then, subtract 38 from both sides to isolate \( l \): \[ l = 62 - 38 \] \[ l = 24 \]

Key Concepts

geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and properties of space. Rectangles are fundamental geometric shapes with opposite sides that are equal in length. A rectangle has two dimensions: length and width. Understanding these basic properties is key to solving rectangle problems. In the case of this exercise, recognizing that the perimeter encompasses the entire outer boundary of the rectangle is crucial.

algebra

Algebra involves working with symbols and the rules for manipulating those symbols. To find the length of a rectangle when given the perimeter and width, we can use algebraic techniques. We start with the perimeter formula and use substitution and solving skills. Here, we were given the perimeter (124) and the width (38), and we needed to find the length. This required manipulating the equation to isolate and solve for the unknown variable (length) by performing basic algebraic operations.

problem-solving steps

Solving any mathematical problem involves a clear understanding of the steps involved.

Here are the steps we followed for the rectangle problem:

• Step 1: Understand the problem and identify the given values. Recognize the formula to use to find the perimeter of the rectangle.
• Step 2: Substitute the given values (perimeter and width) into the formula.
• Step 3: Solve the equation step-by-step to isolate the unknown variable (length). This includes dividing and subtracting as needed.

Each step builds on the previous one, making sure the student's understanding progresses logically.

mathematical formulas

Mathematical formulas are essential tools in problem-solving. The formula for the perimeter of a rectangle is: \( P = 2(l + w) \), where \( P \) is the perimeter, \( l \) is the length, and \( w \) is the width.

By substituting the known values into this formula, we connect geometric properties with algebraic methods to find the unknown value. In this problem, we set up the equation \( 124 = 2(l + 38) \), then isolated \( l \) by dividing both sides by 2 and subtracting 38, leading to \( l = 24 \). Formulas like these illustrate the relationship between different mathematical concepts and help streamline the process of finding solutions.