Determination of Rectangle Dimensions Given Area and Aspect Ratio

Introduction:

When dealing with geometric shapes like rectangles, the dimensions play a crucial role in various calculations. This article will guide you through a problem where the area and the aspect ratio of a rectangle are given, and we need to find the dimensions. Understanding the relationships between these variables is essential for both mathematical and practical applications.

Problem Statement

Consider a rectangle with sides l and L. The area of the rectangle is given by the product of these two sides, A l × L. In this problem, it is specified that one side of the rectangle is twice the length of the other, i.e., L 2 × l. Additionally, the area of the rectangle is given as 100 square units.

Step-by-Step Solution

To solve for the dimensions, let's start with the given information and use algebraic methods:

Area of the rectangle: A l × L Aspect ratio: L 2 × l Area: A 100

Substituting the aspect ratio into the area equation:

A l × L

Since L 2 × l, we can substitute L in the area equation:

A l × (2 × l)

Which simplifies to:

2 × l × l 100

Rewriting it as:

2 × l2 100

Solving for l

Divide both sides of the equation by 2:

l2 50

Take the square root of both sides:

l √50 5√2

Solving for L

Using the relationship L 2 × l, we can now find L by substituting l 5√2 into the equation:

L 2 × 5√2 10√2

Or, we can also express it as:

L √(200)

Conclusion

The dimensions of the rectangle, given the area of 100 square units and the aspect ratio where one side is twice the length of the other, are:

Side l 5√2 Side L 10√2 or √200

By following these steps, we can determine the dimensions of any rectangle given its area and the ratio of its sides. This method is not only useful in mathematical problems but also in various practical applications, such as design and construction.

Key Takeaways:

The relationship between the area and the sides of a rectangle. How to use the aspect ratio to simplify the calculation. The importance of algebraic manipulation in solving geometric problems.

This article provides a clear and concise solution to the given problem, with a step-by-step approach that is easy to follow and understand. Whether you are a student, engineer, or mathematician, this method can be applied to solve similar problems in the future.