How to Find the Area of a Rectangle with Given Perimeter and Length-Width Relationship

In this article, we will explore how to determine the area of a rectangle when given its perimeter and a specific relationship between its length and width. Such problems are common in geometry and are useful in real-world applications, such as designing spaces or planning outdoor areas.

Mathematical Formulation and Problem Statement

A rectangle has a perimeter of 44 units. The length of the rectangle is 5 units less than double the width. Our goal is to find the area of this rectangle.

Solving the Problem

Let's denote the width of the rectangle as ( W ) and the length as ( L ).

Step 1: Express the Relationship Between Length and Width

The length is described as being 5 units less than double the width. This can be expressed as:

[ L 2W - 5 ]

Step 2: Use the Perimeter to Form an Equation

The perimeter of the rectangle is given by the formula:

[ P 2L 2W ]

Since the perimeter is 44, we have:

[ 2L 2W 44 ]

This simplifies to:

[ L W 22 ]

Step 3: Substitute the Length Expression

Substitute ( L 2W - 5 ) into the equation:

[ (2W - 5) W 22 ]

Simplify the equation:

[ 3W - 5 22 ]

Add 5 to both sides:

[ 3W 27 ]

Divide by 3:

[ W 9 ]

Now that we know the width, we can find the length:

[ L 2W - 5 2(9) - 5 18 - 5 13 ]

So the width ( W ) is 9 units and the length ( L ) is 13 units.

Step 4: Calculate the Area

The area ( A ) of the rectangle is given by the formula:

[ A L times W ]

Substituting the values we found:

[ A 13 times 9 117 ]

Therefore, the area of the rectangle is 117 square units.

Conclusion

By following these steps, we were able to determine the area of a rectangle given its perimeter and a relationship between its dimensions. This method can be applied to similar problems in geometry, particularly when dealing with rectangular shapes where the dimensions are related.

Understanding these relationships can be crucial in various fields, such as construction, landscaping, and even everyday problem-solving. If you encounter similar problems, remember to first establish the relationship between the dimensions and then use the perimeter to find those dimensions. Once you have the dimensions, calculating the area becomes straightforward.