Solving for the Dimensions of a Rectangle Given Its Perimeter and Length-Width Relationship

In this article, we will explore the methods to find the dimensions of a rectangle when given the perimeter and the relationship between the length and width. We will walk through step-by-step solutions for various scenarios, ensuring a deep understanding of the problem-solving process. Let's dive in!

Introduction to Rectangles and Perimeter

A rectangle is a quadrilateral with four right angles. The perimeter of a rectangle is the sum of the lengths of its sides. For a rectangle with length L and width W, the perimeter is given by the formula:

Perimeter 2(L W)

Solving for Dimensions

Problem 1: A rectangle has a perimeter of 90 cm. Its length is 5 cm more than its width.

Given:

Perimeter 90 cm Length Width 5 cm

Let's denote the width as W. Then the length L can be written as W 5.

Using the perimeter formula:

2(L W) 90 2(W 5 W) 90 2(2W 5) 90 4W 10 90 4W 80 W 20

Thus, the width is 20 cm and the length is 25 cm.

Problem 2: A rectangle has a perimeter of 70 cm. Its length is 3 cm more than its width.

Given:

Perimeter 70 cm Length Width 3 cm

Let's denote the width as W. Then the length L can be written as W 3.

Using the perimeter formula:

2(L W) 70 2(W 3 W) 70 2(2W 3) 70 4W 6 70 4W 64 W 16

Thus, the width is 16 cm and the length is 19 cm.

Problem 3: A rectangle has a perimeter of 90 and its length is 5 times its width.

Given:

Perimeter 90 Length 5 * Width

Let's denote the width as W. Then the length L can be written as 5W.

Using the perimeter formula:

2(L W) 90 2(5W W) 90 2(6W) 90 12W 90 W 7.5

Thus, the width is 7.5 cm and the length is 37.5 cm.

Problem 4: A rectangle has a perimeter of 10 times its width. Its length is 9 meters more than its width.

Given:

Perimeter 10 * Width Length Width 9 m

Let's denote the width as W. Then the length L can be written as W 9.

Using the perimeter formula:

2(L W) 10W 2(W 9 W) 10W 2(2W 9) 10W 4W 18 10W 18 6W W 3

Thus, the width is 3 meters and the length is 12 meters.

Conclusion

By solving these problems, we have explored different methods to find the dimensions of a rectangle given its perimeter and the relationship between its length and width. These methods involve setting up equations based on the given information and solving for the unknowns. Understanding these steps will help in solving similar problems effectively.

FAQs

What is the perimeter of a rectangle?

The perimeter of a rectangle is the sum of the lengths of all its sides. It is calculated as Perimeter 2(L W), where L is the length and W is the width of the rectangle.

Can the width of a rectangle be negative?

No, the width of a rectangle cannot be negative. A negative width would imply that the rectangle is not properly oriented and does not represent a valid geometric shape. In such cases, the problem setup might be incorrect or the dimensions might need to be reconsidered.

What if the length is a multiple of the width?

If the length is a multiple of the width, it simplifies the problem significantly. You can set up the equation by letting the width be a variable, and the length be a multiple of that variable, then solve for the variable to find the width. From there, the length can be easily determined.