Problem 271 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The length of a rectangle is three meters less than twice the width. The perimeter of the rectangle is 36 meters. Find the dimensions of the rectangle.

Step-by-Step Solution

Answer

The width is 7 meters and the length is 11 meters.

Step 1: Define Variables

Let the width of the rectangle be denoted by \((w)\). The length of the rectangle can be given as three meters less than twice the width, which is \((2w - 3)\).

Step 2: Write the Perimeter Formula

The perimeter of a rectangle is given by the formula: \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width. According to the problem, the perimeter is 36 meters. So, we can write: \(2(l + w) = 36\).

Step 3: Substitute the Length

Substitute the expression for the length into the equation: \((2((2w - 3) + w) = 36)\).

Step 4: Simplify the Equation

Simplify inside the parentheses: \((2w - 3) + w = 3w - 3\) and then multiply by 2: \(2(3w - 3) = 36\). This simplifies to \(6w - 6 = 36\).

Step 5: Solve for the Width

Add 6 to both sides to isolate the term with the variable: \(6w - 6 + 6 = 36 + 6\). This simplifies to: \(6w = 42\). Divide both sides by 6 to solve for \(w\): \(w = 7\) meters.

Step 6: Find the Length

Substitute the value of the width back into the expression for the length: \(l = 2w - 3 = 2(7) - 3 = 14 - 3 = 11\) meters.

Step 7: State the Dimensions

The dimensions of the rectangle are 7 meters for the width and 11 meters for the length.

Key Concepts

Perimeter of a Rectangle

Understanding the perimeter of a rectangle is vital when solving problems involving rectangle dimensions. The perimeter is the total distance around the rectangle. For a rectangle, the formula to calculate the perimeter is given by: \( P = 2(l + w) \), where \(P\) stands for perimeter, \(l\) denotes the length, and \(w\) denotes the width of the rectangle.
In this problem, the given perimeter is 36 meters. By substituting into the perimeter formula: \(2(l + w) = 36\), we can solve for the dimensions of the rectangle.

Variable Substitution

Variable substitution is a technique used to simplify complex equations or expressions by replacing variables with known values or expressions. In this problem:

• Define the width as \(w\).
• The length is three meters less than twice the width, so the expression for length becomes \(2w - 3\).

We substitute this expression into the perimeter formula: \( 2((2w - 3) + w) = 36 \).
Simplifying this creates an equation that only involves the width variable, which we can solve step-by-step.

Linear Equations

Linear equations in one variable can be solved by isolating the variable on one side of the equation. Here, after substituting the length expression into the perimeter formula, we get:

• Simplify inside the parentheses: \(2w - 3 + w = 3w - 3\)
• Multiply by 2: \(2(3w - 3) = 36 \), which expands to \(6w - 6 = 36 \)

To isolate \(w\), add 6 to both sides to get: \(6w = 42 \).
Finally, divide by 6 to find \(w = 7\) meters.
Once the width is known, substitute back to find the length: \(l = 2(7) - 3 = 11 \) meters. The rectangle's dimensions are therefore 7 meters wide and 11 meters long.