Problem 267 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The width of the rectangle is 0.7 meters less than the length. The perimeter of a rectangle is 52.6 meters. Find the dimensions of the rectangle.

Step-by-Step Solution

Answer

Length is 13.5 meters and width is 12.8 meters.

Step 1: Define Variables

Let the length of the rectangle be denoted as \( L \) and the width be denoted as \( W \).

Step 2: Set Up Relationships

According to the problem, the width of the rectangle is 0.7 meters less than the length. Therefore, we can write \( W = L - 0.7 \).

Step 3: Use Perimeter Formula

The formula for the perimeter of a rectangle is \( P = 2L + 2W \). Substitute the given perimeter, 52.6 meters: \( 52.6 = 2L + 2W \).

Step 4: Substitute Width Expression

Substitute \( W \) from Step 2 into the perimeter formula: \( 52.6 = 2L + 2(L - 0.7) \).

Step 5: Simplify the Equation

Simplify and solve for \( L \): \( 52.6 = 2L + 2L - 1.4 \) therefore, \( 52.6 = 4L - 1.4 \), and \( 54 = 4L \).

Step 6: Solve for Length

Divide both sides by 4 to find \( L \): \( L = 13.5 \).

Step 7: Solve for Width

Use the relationship between length and width: \( W = L - 0.7 \) therefore, \( W = 13.5 - 0.7 = 12.8 \).

Key Concepts

rectangle properties

Understanding rectangle properties is crucial to solving these types of exercises. A rectangle is a four-sided figure where opposite sides are equal in length. It has four right angles, which makes it easier to apply certain formulas. For example, knowing the relationship between length and width helps us set up an equation.
With these properties in mind, we can begin solving for dimensions by defining variables for length and width. This step simplifies the process and lays the groundwork for further calculations.

perimeter calculations

The formula for the perimeter of a rectangle is essential: it is the sum of all sides. For a rectangle, this is expressed as: \[ P = 2L + 2W \]
When given the perimeter, we can rearrange this formula to find either the length or the width.

• First, substitute the given perimeter value into the equation.
• Use any relationships between length and width, like those provided in the problem, to simplify the equation further.

This process makes it easier to solve the equation for the unknown variables.

algebraic substitution

Algebraic substitution is sliding known values or expressions into another equation to simplify and solve it. In this exercise, we use the relationship between width and length: \[ W = L - 0.7 \]
By substituting this into the perimeter equation, we transform it into: \[ 52.6 = 2L + 2(L - 0.7) \]
This reduces the number of variables, making the equation easier to manage. Afterwards, distribute and combine like terms for simplicity before solving for the unknown variable.

solving equations

Solving equations often involves steps like isolation, substitution, and simplifying expressions. After substitution, we get the equation: \[ 52.6 = 2L + 2L - 1.4 \]
Combining like terms gives: \[ 52.6 = 4L - 1.4 \]
Adjust the equation by adding 1.4 to both sides: \[ 54 = 4L \]
Finally, divide both sides by 4 to isolate the length: \[ L = 13.5 \]
Using the previously established relationship, the width is found by: \[ W = 13.5 - 0.7 = 12.8 \]
This step-by-step approach ensures each part of the equation is clear and manageable.