Problem 268 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The length of the rectangle is 1.1 meters less than the width. The perimeter of a rectangle is 49.4 meters. Find the dimensions of the rectangle.

Step-by-Step Solution

Answer

Width = 12.9 meters; Length = 11.8 meters.

Step 1 - Define the Variables

Let the width of the rectangle be denoted as \(w\), and the length of the rectangle be denoted as \(l\). According to the problem, the length (\(l\)) is \(1.1\) meters less than the width (\(w\)). Thus, we write: \[l = w - 1.1\]

Step 2 - Write the Perimeter Formula

The perimeter \(P\) of a rectangle is given by the formula: \[P = 2l + 2w\] According to the problem, the perimeter of the rectangle is \(49.4\) meters. Therefore, we can write: \[2l + 2w = 49.4\]

Step 3 - Substitute the Length in Perimeter Formula

Substitute \(l = w - 1.1\) into the perimeter formula: \[2(w - 1.1) + 2w = 49.4\]

Step 4 - Simplify and Solve for Width

Distribute and combine like terms to solve for \(w\): \[\begin{align*} 2w - 2.2 + 2w &= 49.4 \ 4w - 2.2 &= 49.4 \ 4w &= 51.6 \ w &= \frac{51.6}{4} \ w &= 12.9 \text{ meters} \ \text{Therefore, the width } w = 12.9 \text{ meters} \ \text{and } l = w - 1.1 = 12.9 - 1.1 = 11.8 \text{ meters.} \ \text{So the length } l = 11.8 \text{ meters.} \ \text{The two dimensions of the rectangle are } l = 11.8 \text{ meters and } w = 12.9 \text{ meters. }\ \end{align*}\]

Key Concepts

Rectangle Properties

To start solving this problem, it's important to understand some basic properties of rectangles. A rectangle has four sides with opposite sides being equal in length. The two longer sides are called the length (often denoted as \( l \)), and the two shorter sides are the width (often denoted as \( w \)).
Since opposite sides are equal, this gives rise to several useful formulas such as those for its perimeter and area. These foundational properties will help us solve more complex problems involving rectangles.

Perimeter Formula

The perimeter of a rectangle is the total distance around the edge of the rectangle. To find the perimeter, you add up the lengths of all four sides. Since a rectangle has two lengths and two widths, its perimeter \( P \) can be found using the formula:

\[ P = 2l + 2w \]

This formula is very useful in problems involving the dimensions of rectangles, as it links the perimeter directly to both the length and the width of the shape.

Algebraic Equations

In many rectangle problems, algebra is used to find unknown dimensions. This involves creating and solving equations. In our problem, we use the relationship that the length \( l \) is 1.1 meters less than the width \( w \). So, we write:

\( l = w - 1.1 \)

Next, we substitute this into the perimeter formula to form an algebraic equation. Knowing the perimeter helps us convert a geometric property into a solvable math problem. By setting up and solving these equations, we can determine the dimensions of the rectangle.

Step-by-Step Solution

Let's summarize the step-by-step solution to our problem.

First, we define our variables. Let \( w \) be the width and \( l \) be the length. Given that \( l = w - 1.1 \), we set up the perimeter equation:

\[ 2l + 2w = 49.4 \]

Next, we substitute \( l = w - 1.1 \) into the perimeter equation:

\[ 2(w - 1.1) + 2w = 49.4 \]

We then distribute and combine like terms:

\[ 2w - 2.2 + 2w = 49.4 \]

\[ 4w - 2.2 = 49.4 \]

Solving for \( w \):

\[ 4w = 51.6 \]

\[ w = \frac{51.6}{4} \]

\[ w = 12.9 \text{ meters} \]

Finally, we find the length by substitution:

\[ l = 12.9 - 1.1 = 11.8 \text{ meters} \]

So, the dimensions of the rectangle are \(l = 11.8\) meters and \(w = 12.9\) meters.