Problem 30 - Algebra 1: Concepts and Skills

Question

The rectangle at the right has a perimeter of 40 centimeters. The length of the rectangle is 4 times as long as the width. Find the dimensions of the rectangle.

Step-by-Step Solution

Answer

The dimensions of the rectangle are length 16 cm and width 4 cm.

Step 1: Setup

Write down the formula for the perimeter of a rectangle. P = 2l + 2w.

Step 2: Insert the given information

Substitute the given information into the perimeter formula. We know that P = 40 cm, l = 4w, so our equation becomes: 40 = 2(4w) + 2w.

Step 3: Simplify the equation

Simplify the equation: 40 = 8w + 2w which results in 40 = 10w

Step 4: Solve for width

Solve the equation for w (width of the rectangle) by dividing both sides by 10: w = 4 cm

Step 5: Solve for length

Substitute w = 4 cm into l = 4w to find the length of the rectangle: l = 4*4 cm = 16 cm

Key Concepts

Rectangle Dimensions

To find the dimensions of a rectangle, you need to understand the relationship between its perimeter, length, and width. A rectangle has two pairs of equal sides - the length and the width.

• The **perimeter** of a rectangle is the total length around the shape. It is calculated by the formula: \[ P = 2l + 2w \] where \( P \) is the perimeter, \( l \) is the length, and \( w \) is the width.
• In our problem, the perimeter is given as 40 cm. We also know the length is 4 times the width, expressed as \( l = 4w \).

Understanding these expressions and how they relate to each other is key to determining the sides of the rectangle.

Solving Linear Equations

Linear equations contain variables (like \( w \) for width in our example) and involve simple algebraic operations. The aim is to solve for one variable by isolating it on one side of the equation.

• Start by substituting known values into the given formula. In this case, the perimeter formula becomes \( 40 = 2(4w) + 2w \) after inserting \( l = 4w \).
• Next, simplify the equation to combine like terms, which gives us \( 40 = 8w + 2w \).
• Combine the \( w \) terms to streamline the equation: \( 40 = 10w \).

This linear equation is now ready to solve, aiming to find the precise width value.

Substitution Method

The substitution method is a useful technique, especially when one equation defines a variable in terms of another. Here’s how it works:

• In our problem, we use substitution to express the length in terms of the width: \( l = 4w \).
• After calculating \( w \) using the simplified equation \( 40 = 10w \), we find \( w = 4 \) cm.
• Next, substitute the value of \( w \) back into \( l = 4w \) to determine the length: \( l = 4 \times 4 \) cm, resulting in \( l = 16 \) cm.

This method helps in breaking down the process into manageable steps, making it easier to find unknown dimensions using a given relationship.