Problem 270 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The length of a rectangle is three times the width. The perimeter of the rectangle is 72 feet. Find the length and width of the rectangle.

Step-by-Step Solution

Answer

The width is 9 feet and the length is 27 feet.

Step 1: Define Variables

Let the width of the rectangle be denoted as \(w\). Since the length is three times the width, let the length be denoted as \(3w\).

Step 2: Write Perimeter Formula

The perimeter of a rectangle is calculated using the formula: \[ P = 2 \times (\text{Length} + \text{Width}) \]Here, the perimeter \(P\) is given as 72 feet.

Step 3: Substitute Values into the Formula

Substitute \(3w\) for the length and \(w\) for the width into the perimeter formula:\[ 72 = 2 \times (3w + w) \]

Step 4: Simplify the Equation

Combine like terms inside the parentheses:\[ 72 = 2 \times 4w \]Then, simplify further:\[ 72 = 8w \]

Step 5: Solve for the Width

Divide both sides of the equation by 8 to solve for \(w\):\[ w = \frac{72}{8} \]\[ w = 9 \]

Step 6: Find the Length

Use the value of \(w\) to find the length. Since the length is three times the width:\[ 3w = 3 \times 9 = 27 \]

Key Concepts

rectangle properties

When working with rectangle problems in algebra, it's important to understand the basic properties of a rectangle. A rectangle has four sides and opposite sides are equal in length. This means if you know the length of one side, you can easily determine the length of the opposite side. Also, a rectangle has four right angles, each being 90 degrees. Remembering these properties can help simplify many problems. Here, we were given that the length is three times the width, showing a specific proportional relationship which is common in area and perimeter problems.

perimeter calculation

The perimeter of a rectangle can be calculated using a straightforward formula. The formula is: \( P = 2 \times (\text{Length} + \text{Width}) \). This formula adds the lengths of all four sides of the rectangle together. For instance, in our problem, the perimeter is 72 feet. By substituting the given perimeter value and the expressions for the length and width, we can create an equation that models the problem. This step is crucial for setting up our problem for solution.

variable substitution

Variable substitution is a powerful tool in algebra. It involves replacing variables with their known values. In our problem, we first defined the width of the rectangle as \( w \), and since the length is three times the width, we let the length be \( 3w \). Substitution simplifies complex expressions and allows us to work with simpler equations. This is especially useful when dealing with formulas and equations where multiple variables are involved. Substituting \( 3w \) and \( w \) into the perimeter formula is a key step in solving for the unknown dimensions.

solving linear equations

Solving linear equations involves isolating the variable to find its true value. After substituting our variables into the perimeter formula, our equation became: \( 72 = 2 \times (3w + w) \), which simplifies to \( 72 = 8w \). To solve for the width \( w \), we divide both sides of the equation by 8, giving \( w = 9 \). Once we know \( w \), we can find the length by calculating \( 3w = 27 \). Step-by-step simplification and solving for the variable is fundamental in algebra, ensuring every step is logical and clear.