Problem 11 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. The length of a rectangle is twice its width. If the perimeter of the rectangle is 28 in., what are its dimensions?

Step-by-Step Solution

Answer

Width: 4.67 inches, Length: 9.34 inches

Step 1: Define Variables

Let the width of the rectangle be denoted as \( w \). According to the problem, the length of the rectangle is twice its width, so let the length be denoted as \( 2w \).

Step 2: Write the Perimeter Formula

The perimeter of a rectangle is calculated with the formula: \[ P = 2 \times (\text{length} + \text{width}) \] Given the perimeter is 28 inches, substitute the expressions for length and width: \[ 28 = 2 \times (2w + w) \]

Step 3: Simplify the Equation

Combine like terms inside the parentheses: \[ 28 = 2 \times (3w) \]Then multiply by 2: \[ 28 = 6w \]

Step 4: Solve for Width

To find the width, solve for \( w \): \[ w = \frac{28}{6} = \frac{14}{3} = 4.67 \text{ inches} \]

Step 5: Find the Length

Now use the width to find the length. Since the length is twice the width: \[ \text{Length} = 2w = 2 \times 4.67 = 9.34 \text{ inches} \]

Key Concepts

Rectangle Dimensions

In this exercise, we are finding the dimensions of a rectangle, specifically its length and width. A rectangle has two pairs of parallel sides. The length is the longer side, and the width is the shorter side.

To find the dimensions of a rectangle algebraically, follow these steps:

• Define variables for width and length.
• Use the information given to create equations.

For example, if the length is twice the width, you can set up your variables as follows:

• Let the width be denoted as \(w\).
• Then, the length will be \(2w\).

This setup is crucial to solve algebraic word problems involving rectangles.

Perimeter Formula

The perimeter of a rectangle is the total distance around the edges of the rectangle. The formula for the perimeter \(P\) of a rectangle is: $$$ P = 2 \times (\text{length} + \text{width}) $$$.

Here's what each part of the formula means:

• The \(2 \times\) accounts for both pairs of sides.
• The sum \( \text{length} + \text{width} \) adds the two different side lengths.

Using this formula, you can substitute the values from the problem. For instance, if the perimeter is 28 inches and the length is twice the width, you can write:

\[ 28 = 2 \times (2w + w) \]
You then simplify and solve this equation to find the actual measurements.

Solving Equations

Solving equations is a fundamental part of algebra. Here, we solve for the width and length of the rectangle by setting up and simplifying the perimeter equation.

Let's look at the key steps:

• Firstly, substitute the given dimensions into the perimeter formula: \[ 28 = 2 \times (2w + w) \]
• Next, combine like terms: \[ 28 = 2 \times 3w \]
• Multiply to simplify: \[ 28 = 6w \]
• Solve for the variable: \[ w = \frac{28}{6} = \frac{14}{3} = 4.67 \text{ inches} \]

With the width found, you can then determine the length using the relationship given in the problem: \[ \text{Length} = 2w = 2 \times 4.67 = 9.34 \text{ inches} \]
Solving equations step by step ensures that you get accurate results. Always perform one step at a time.