When comparing the areas of a rectangle and a square, the relationship between their perimeters can provide fascinating insights. This article delves into the scenario where the perimeters of a rectangle and a square are equal, and the sides of the rectangle are in a 2:3 ratio. We will determine the ratio of the areas of these two shapes, utilizing the principles of geometry and algebra.

Introduction

In geometry, the perimeter and area of shapes are fundamental concepts used in various applications, from architecture to mathematics. This article focuses on the specific case where a rectangle and a square have the same perimeter, and explores how their side lengths and areas are related. Understanding such relationships can be particularly useful in optimization problems and design-related scenarios.

Problem Setup

The problem statement provides us with the following information:

The perimeter of a rectangle is 22x3. Two adjacent sides of the rectangle are in a 2:3 ratio. The perimeter of a square is 1.

To solve this problem, we need to determine the areas of both the rectangle and the square and then find the ratio of these areas.

Step-by-Step Solution

1. Determine the Side Lengths of the Rectangle

Let the sides of the rectangle be 2x and 3x. The perimeter of the rectangle is given as 22x3. We can set up the perimeter equation as follows:

2(2x 3x) 22x3

2(5x) 22x3

1 22x3

x 22/10 2.2

With x determined, we can find the side lengths of the rectangle:

First adjacent side: 2x 2(2.2) 4.4 Second adjacent side: 3x 3(2.2) 6.6

The area of the rectangle can be calculated as:

Area of the rectangle 4.4 * 6.6 28.96 square units

2. Determine the Side Length of the Square

The perimeter of the square is given as 1. We can set up the perimeter equation for the square as follows:

4s 1

s 1/4 0.25

The side length of the square is 0.25.

The area of the square can be calculated as:

Area of the square (0.25)^2 0.0625 square units

3. Calculate the Ratio of the Areas

The ratio of the area of the rectangle to the area of the square is calculated as:

Ratio Area of the rectangle / Area of the square

Ratio 28.96 / 0.0625 463.36

However, to express this in a simplified numerical ratio, we can convert it back to a simpler form. If we use the original variable x instead of the numerical values, the areas become:

Area of the rectangle 6x^2

Area of the square 2.5x^2 6.25x^2

The ratio of the areas is then:

Ratio 6x^2 / 6.25x^2 24/25

Conclusion

When the perimeters of a rectangle and a square are equal, and the sides of the rectangle are in a 2:3 ratio, the ratio of the area of the rectangle to the area of the square is 24:25. This calculation highlights the importance of understanding the relationship between side lengths, perimeters, and areas in geometric shapes, which can be applied in a variety of mathematical and practical scenarios.