Problem 22 - Fundamentals of Mathematics

Question

The perimeter of a rectangle is 18 meters. Find the length and width of the rectangle if the length is 1 meter more than three times the width.

Step-by-Step Solution

Answer

Width is 2 meters, and length is 7 meters.

Step 1: Understanding the Problem

We are given that the perimeter of a rectangle is 18 meters. We also know that the length of the rectangle is 1 meter more than three times its width. We need to find the actual measurements of the length and width.

Step 2: Define Variables

Let's define the width of the rectangle as \( w \) meters. Based on the information provided, the length \( l \) would be \( 3w + 1 \) meters because the length is 1 more than three times the width.

Step 3: Perimeter Formula

The formula for the perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \). We know that \( P = 18 \) meters in this problem.

Step 4: Set Up the Equation

Substitute the expressions for \( l \) and \( P \) into the perimeter formula: \[ 2((3w + 1) + w) = 18 \]

Step 5: Simplify the Equation

Simplify the expression inside the parentheses first:\[ 2(3w + 1 + w) = 18 \] which becomes \[ 2(4w + 1) = 18 \]

Step 6: Solve for Width

Now distribute the 2:\[ 8w + 2 = 18 \] Subtract 2 from both sides to isolate the term with \( w \):\[ 8w = 16 \] Divide both sides by 8 to solve for \( w \):\[ w = 2 \] meters.

Step 7: Calculate the Length

Now that we have \( w = 2 \) meters, use the expression for the length to find it:\[ l = 3w + 1 = 3(2) + 1 = 7 \] meters.

Key Concepts

Rectangle Properties

A rectangle is a four-sided polygon with opposite sides that are equal and parallel. Understanding the basic properties of rectangles helps lay the groundwork for solving many mathematical problems, including perimeter calculations.

Key properties of rectangles include:

• Opposite sides are of equal length.
• Each angle in a rectangle is a right angle, meaning each is exactly 90 degrees.
• Rectangles have two diagonals that are equal in length.
• The perimeter and area are two primary properties used to describe the size of a rectangle.

By understanding these properties, we can efficiently set up equations to solve for unknown dimensions such as length and width when certain measurements, like the perimeter, are provided.

Perimeter Formula

The perimeter of a rectangle is simply the total distance around the shape. This linear measurement is crucial when solving problems related to border, fencing, or any other application involving edge length.

The conventional formula for the perimeter \( P \) of a rectangle involves adding the lengths of all four sides, which is given by:

• \( P = 2(l + w) \)

where \( l \) represents the length and \( w \) represents the width. In our problem, we know the perimeter is 18 meters. Applying this formula, we set up the equation to find the unknown variables. Recognizing how the formula incorporates both length and width is essential in using it to solve perimeter problems.

Algebraic Equations

In solving rectangular problems, algebraic equations act as our best friend, providing a systematic approach to finding unknown values. Solving these problems generally involves setting up an equation based on a given formula.Given our rectangle problem, we defined the width as \( w \) and the length in terms of this width as \( l = 3w + 1 \). By substituting these expressions into the perimeter formula \( 2(l + w) = 18 \), we set up the equation:\[ 2((3w + 1) + w) = 18 \]This equation is simplified and solved to find \( w \). The art of dealing with these equations involves careful balancing and understanding linear relationships to isolate and solve for unknowns. After isolating \( w \), we can directly substitute back to calculate the length, effectively utilizing algebra to derive the needed dimensions. By manipulating the equation step by step, students can manage similar problems with confidence.