Question

Geometry The perimeter of a rectangle is 42 meters. Find its length and width if the length is twice the width.

Short answer

Length: 14 meters, Width: 7 meters

Step-by-step solution

Understand the problem

The problem provides the perimeter of a rectangle as 42 meters and states that the length is twice the width. The goal is to find the dimensions of the rectangle.

State the formulas

Use the perimeter formula for a rectangle: \[ P = 2L + 2W \] Here, \( P \) is the perimeter, \( L \) is the length, and \( W \) is the width. Also, note that the length is twice the width: \[ L = 2W \]

Substitute the length value

Replace \( L \) in the perimeter formula with \( 2W \): \[ P = 2(2W) + 2W \] Given that \( P = 42 \), substitute 42 for \( P \): \[ 42 = 4W + 2W \]

Simplify and solve for width

Combine the terms with \( W \): \[ 42 = 6W \]Solve for \( W \) by dividing both sides by 6: \[ W = 7 \text{ meters} \]

Find the length

Since the length is twice the width, use the equation \( L = 2W \): \[ L = 2(7) = 14 \text{ meters} \]

Verify the solution

Verify by plugging the found values back into the perimeter formula: \[ P = 2L + 2W = 2(14) + 2(7) = 28 + 14 = 42 \text{ meters} \]The perimeter matches the given value, so the solution is correct.

Key concepts

Perimeter Formula

In geometry, the perimeter of a shape is the total distance around the edge of the shape. For a rectangle, you can use a specific perimeter formula. The formula is \ P = 2L + 2W \ where \( P \) stands for the perimeter, \( L \) represents the length, and \( W \) is the width. Essentially, this formula adds up all the sides of the rectangle. Because a rectangle has two pairs of equal sides, each length and width is counted twice. Understanding this formula is crucial for solving problems involving rectangle dimensions.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. These equations often help us find unknown values. In the context of our rectangle problem, we had to find the width \( W \) given a certain perimeter and relationship between length and width. Starting with the perimeter formula \ P = 2L + 2W \ and given \( P = 42 \) meters, we substituted \( L \) with \( 2W \) because the length is twice the width: \ 42 = 2(2W) + 2W. \We then simplified this to \ 42= 4W + 2W. \ This is a linear equation in terms of \( W \). By combining like terms, we get \ 42 = 6W, \ and by dividing both sides by 6 we solve for \( W \): \ W = 7. Knowing how to isolate variables and solve these types of equations is essential in algebra.

Rectangle Dimensions

To find the dimensions of a rectangle, you need both its length and width. In our problem, we found the width first using the provided information. Once we solved for the width as \( 7 \) meters, we used the relationship that the length is twice the width. This gave us the length as \( L = 2W = 2(7) = 14 \) meters. Verifying these dimensions ensures they satisfy the perimeter formula: \ P = 2L + 2W = 2(14) + 2(7) = 28 + 14 = 42 \ meters. This step confirms we correctly identified the rectangle's dimensions. Mastering this process of confirmation is key to solving geometry problems.