Problem 18 - Elementary Algebra

Question

The width of a rectangle is one-half of its length. If the perimeter of the rectangle is 54 feet, find its length and width.

Step-by-Step Solution

Answer

The length is 18 feet and the width is 9 feet.

Step 1: Understand the formula for perimeter

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

Step 2: Establish the relationship between length and width

We are given that the width \( W \) is one-half of the length \( L \). Thus, we can express \( W \) as:\[ W = \frac{1}{2}L \]

Step 3: Substitute the width in the perimeter formula

Substitute \( W = \frac{1}{2}L \) into the perimeter formula:\[ P = 2L + 2\left(\frac{1}{2}L\right) \] This simplifies to:\[ P = 2L + L \] or\[ P = 3L \]

Step 4: Solve for the length

With the given perimeter \( P = 54 \) feet, substitute into the equation:\[ 54 = 3L \] Now solve for \( L \) by dividing both sides by 3:\[ L = \frac{54}{3} = 18 \] Thus, the length \( L \) is 18 feet.

Step 5: Find the width using the relationship

Use the relationship \( W = \frac{1}{2}L \) to find the width:\[ W = \frac{1}{2} \times 18 = 9 \] Thus, the width \( W \) is 9 feet.

Key Concepts

Length and Width Relationship

Understanding the relationship between the length and width of a rectangle is essential for solving problems involving perimeter. In this case, the width is defined as being one-half of its length. This type of relationship is linear, meaning you can directly compute one measurement knowing the other.

Imagine a rectangle: if the length is longer, the width will automatically adjust to half of that longer measure. So, if the length (\( L \)) is 10 feet, the width (\( W \)) is 5 feet since \( W = \frac{1}{2}L \).

The understanding of such relationships allows you to create equations that represent these ratios or comparisons, helping solve numerous geometric problems easily.

• This recognition saves time and efforts, particularly in solving for multiple parameters.
• It allows the conversion of word problems into mathematical expressions, where each variable has a clear role.

Perimeter Formula

The perimeter of a rectangle is the total distance around the edge of the figure. This can be calculated using the formula: \[ P = 2L + 2W \] where \( P \) represents the perimeter, \( L \) is the length, and \( W \) is the width. This formula stems from the rectangle having two lengths and two widths which sum up to create the perimeter measurement.

This formula is especially useful because even if you know only one of the sides, you can use information about the perimeter to solve for the unknown dimensions.

• It is applicable in practical scenarios like fencing a yard or outlining a room.
• Substituting known values makes it straightforward to calculate unknowns, enhancing problem-solving efficacy.

When you integrate known relationships (like \( W = \frac{1}{2}L \)), this formula becomes a crucial part of seamlessly linking other calculations with the geometric reality.

Solving Linear Equations

Linear equations serve as a powerful tool in solving geometric problems involving relationships. In this rectangle problem, once you establish the relationship \( W = \frac{1}{2}L \) and plug it into the perimeter formula, it simplifies to \( 3L = P \).

This equation is linear as it forms a straight line on a graph, and it can be solved using basic operations—addition, subtraction, multiplication, and division.

• For example, to solve \( 54 = 3L \), divide both sides by 3, obtaining \( L = 18 \).
• Once you have \( L \), substitute it back into any relational expressions, like \( W = \frac{1}{2}L \), to find the width.

Grasping how to manipulate and solve linear equations is key not only in math but also in everyday decision-making where proportions and balanced relationships arise.

Linear equation solving fosters a logical approach to complex problems, turning broader challenges into manageable steps and solutions.