Problem 31 - Elementary Algebra

Question

The length of a rectangle is 6 meters more than three times the width. The perimeter of the rectangle is 44 meters What are the dimensions of the rectangle?

Step-by-Step Solution

Answer

Answer: The dimensions of the rectangle are 18 meters in length and 4 meters in width.

Step 1: Set up the formula for the perimeter of a rectangle

The formula for the perimeter P of a rectangle is given by: P = 2(L + W), where L is the length, and W is the width.

Step 2: Identify the given information

We are given that the perimeter of the rectangle is 44 meters, and the length is 6 meters more than three times the width (L = 3W + 6).

Step 3: Substitute the given information into the formula

Substitute the given information into the formula: 44 = 2(3W + 6 + W)

Step 4: Simplify the equation

Simplify the equation to solve for the width: 44 = 2(4W + 6)

Step 5: Distribute and solve for W

Distribute the 2 on the right side of the equation: 44 = 8W + 12. Now, solve for the width: 8W = 32 W = 4 meters

Step 6: Calculate the length

Now that we have the width, we can find the length using the relationship L = 3W + 6. Substitute the width (W = 4 meters) into the relationship: L = 3(4) + 6 L = 12 + 6 L = 18 meters

Step 7: State the final dimensions

The dimensions of the rectangle are 18 meters in length and 4 meters in width.

Key Concepts

Perimeter of a Rectangle

When dealing with rectangle-related problems in algebra, one of the key aspects to understand is the perimeter. Perimeter refers to the total distance around the edges of any polygon. For rectangles, the perimeter is calculated using the formula:

• \( P = 2(L + W) \)

Where:

• \(L\) is the length
• \(W\) is the width

The perimeter formula sums up twice the length and twice the width. In simple terms, it is like adding up the sides of the rectangle going around it. This formula is vital in various problems, whether you are given the perimeter and need to find a missing dimension, or you have the dimensions and need to find the perimeter. Understanding how to manipulate this formula is foundational in algebra word problems relating to geometry.

Solving Equations

Solving equations is a fundamental skill in algebra, allowing us to find unknown values. In the word problem given, you're working with an equation derived from the perimeter formula:

• \( 44 = 2(3W + 6 + W) \)

To solve equations like this, follow these steps:

• Substitute known values into the formula.
• Simplify the equation by combining like terms and distributing constants.
• Isolate the variable (in this case, \(W\) for width) by performing inverse operations.

For this problem, you simplify and solve for \(W\) to find the width of the rectangle. Once solved, the width becomes a known value, which is then used to find other unknowns like the length. Mastering these techniques is crucial in not just geometry, but across all algebraic problems.

Dimensions of a Rectangle

Finding the dimensions of a rectangle often involves relationships between its length and width. In our given problem, the relationship is described as:

• Length \(L\) is 6 meters more than three times the width \(W\).

This can be written as the equation:

• \(L = 3W + 6\)

Once you have the width from solving the perimeter equation, it is straightforward to find the length using this relationship. Simply substitute the known width back into the equation to calculate the length. This step-by-step use of relationships within the rectangle helps cement the understanding of how algebra connects various pieces of information to solve geometry problems. It's an excellent illustration of the power of algebra in making calculations clear and easily manageable.