Dimensions Of Mr. Chaudhary's Pond Finding Length, Breadth, And Percentage Increase For A Square Shape
Introduction: Delving into the Geometry of Fish Farming
Mr. Chaudhary, an astute aquaculturist, possesses a rectangular pond dedicated to fish farming. This aquatic haven sprawls across 8000 square meters of his land, its perimeter stretching a considerable 360 meters. Our task is twofold: first, to unravel the dimensions of this pond, specifically its length and breadth; and second, to determine the percentage by which the shorter edge must be increased to transform the rectangle into a square. This exploration delves into the practical application of mathematical principles, specifically the concepts of area and perimeter, in a real-world scenario. Understanding these concepts is not only crucial for solving this particular problem but also for gaining a broader appreciation of how geometry shapes our everyday lives.
At the heart of this problem lies the interplay between area and perimeter, two fundamental properties of two-dimensional shapes. The area of a rectangle, the space it occupies, is calculated by multiplying its length and breadth. The perimeter, on the other hand, is the total distance around the rectangle, found by adding up the lengths of all its sides. In Mr. Chaudhary's case, we are given both the area and the perimeter of the pond, providing us with a system of equations that we can solve to find the unknown length and breadth. This process exemplifies how mathematical equations can be used to model and solve real-world problems, allowing us to quantify and understand the physical world around us. The journey from abstract equations to concrete dimensions highlights the power of mathematical reasoning and its practical applications.
Moreover, this problem introduces the concept of geometric transformation, specifically the transformation of a rectangle into a square. A square, a special type of rectangle, has all its sides equal in length. To transform Mr. Chaudhary's rectangular pond into a square, we need to adjust its dimensions. The question asks us to determine the percentage increase required for the shorter edge to match the length of the longer edge. This involves understanding the relationship between the sides of a rectangle and a square, and how changes in one dimension affect the overall shape. This exploration not only reinforces geometric principles but also introduces the idea of optimization, a crucial concept in various fields, from engineering to economics. By understanding how to manipulate shapes and sizes, we can optimize designs and processes to achieve desired outcomes.
(i) Unraveling the Dimensions: Finding Length and Breadth
To embark on our quest to find the length and breadth of Mr. Chaudhary's pond, we must first translate the given information into a mathematical framework. We know the area of the rectangular pond is 8000 square meters, and its perimeter is 360 meters. Let's denote the length of the pond as 'l' and the breadth as 'b'. With these variables in hand, we can express the area and perimeter using mathematical equations. The area of a rectangle is given by the product of its length and breadth, so we have the equation: l * b = 8000. The perimeter of a rectangle is given by twice the sum of its length and breadth, leading to the equation: 2 * (l + b) = 360. These two equations form a system of equations that we can solve to determine the values of 'l' and 'b'.
The next step involves employing algebraic techniques to solve the system of equations. We have two equations and two unknowns, a classic scenario in algebra. One common method is substitution. From the perimeter equation, we can isolate the sum of the length and breadth: l + b = 180. Then, we can express one variable in terms of the other. For instance, we can write b = 180 - l. Now, we can substitute this expression for 'b' into the area equation: l * (180 - l) = 8000. This substitution transforms the system of equations into a single quadratic equation in one variable, 'l'. Solving this quadratic equation will give us the possible values for the length of the pond. The process of substitution is a powerful tool in algebra, allowing us to simplify complex systems of equations and find solutions.
The quadratic equation we obtained is: 180l - l^2 = 8000. Rearranging the terms, we get: l^2 - 180l + 8000 = 0. To solve this quadratic equation, we can use the quadratic formula, a general formula for finding the roots of any quadratic equation of the form ax^2 + bx + c = 0. The quadratic formula is given by: x = [-b ± sqrt(b^2 - 4ac)] / 2a. In our case, a = 1, b = -180, and c = 8000. Plugging these values into the formula, we get: l = [180 ± sqrt((-180)^2 - 4 * 1 * 8000)] / 2 * 1. Simplifying this expression will give us the two possible values for the length 'l'. These values represent the solutions to the quadratic equation and, consequently, the possible lengths of Mr. Chaudhary's pond. The quadratic formula is a cornerstone of algebra, providing a reliable method for solving quadratic equations and unlocking solutions to a wide range of problems.
After applying the quadratic formula and simplifying, we obtain two possible values for the length: l = 100 meters and l = 80 meters. Corresponding to these values, we can find the breadth using the equation b = 180 - l. If l = 100 meters, then b = 180 - 100 = 80 meters. Conversely, if l = 80 meters, then b = 180 - 80 = 100 meters. Therefore, the dimensions of the pond are either length = 100 meters and breadth = 80 meters, or length = 80 meters and breadth = 100 meters. Since length is typically considered the longer side, we can conclude that the length of the pond is 100 meters and the breadth is 80 meters. This result provides a concrete answer to the first part of our problem, revealing the physical dimensions of Mr. Chaudhary's fish pond.
(ii) Transforming the Rectangle: Percent Increase for a Square
Now, we turn our attention to the second part of the problem: determining the percentage by which the shorter edge of the pond must be increased to transform it into a square. We know that Mr. Chaudhary's pond is currently a rectangle with a length of 100 meters and a breadth of 80 meters. To transform this rectangle into a square, we need to make all sides equal in length. This means the shorter side, the breadth, needs to be increased to match the length. The core concept here is understanding the properties of a square and how it differs from a rectangle. A square is a special type of rectangle where all four sides are of equal length. The transformation we are considering involves altering the dimensions of the rectangle to achieve this equality.
To calculate the required increase, we first need to determine the difference between the length and the breadth. The length of the pond is 100 meters, and the breadth is 80 meters, so the difference is 100 - 80 = 20 meters. This means the breadth needs to be increased by 20 meters to match the length and form a square. This difference represents the absolute amount of increase required. However, the question asks for the percentage increase, which expresses the increase as a proportion of the original breadth. Understanding the concept of percentage is crucial here, as it allows us to express the increase in a relative manner, making it easier to compare with other scenarios.
To calculate the percentage increase, we divide the amount of increase by the original breadth and then multiply by 100. The amount of increase is 20 meters, and the original breadth is 80 meters. Therefore, the percentage increase is (20 / 80) * 100. Simplifying this expression, we get (1/4) * 100 = 25%. This calculation reveals the percentage by which the shorter edge must be increased to transform the rectangle into a square. The concept of percentage increase is widely used in various fields, from finance to statistics, to express changes in a standardized way.
Therefore, the shorter edge of the pond, the breadth, must be increased by 25% to make it a square. This means that if we increase the breadth of the pond by 25%, it will become equal to the length, and the pond will transform into a square shape. This result provides a clear and concise answer to the second part of our problem, demonstrating the application of percentage calculations in geometric transformations. By understanding the relationship between the sides of a rectangle and a square, and by applying the concept of percentage increase, we can effectively manipulate shapes and sizes to achieve desired outcomes. This ability to transform shapes has numerous practical applications, from designing structures to optimizing layouts.
Conclusion: Mathematical Insights into Aquaculture
In conclusion, we have successfully navigated the geometric landscape of Mr. Chaudhary's fish pond, uncovering its dimensions and determining the transformation required to reshape it into a square. Through the application of mathematical principles, we found that the pond has a length of 100 meters and a breadth of 80 meters. Furthermore, we calculated that the shorter edge, the breadth, needs to be increased by 25% to achieve a square shape. This exploration highlights the practical relevance of mathematics in real-world scenarios, particularly in fields like aquaculture, where understanding spatial relationships and geometric transformations can be crucial for efficient design and resource management.
This problem not only provided an opportunity to apply mathematical concepts but also showcased the power of problem-solving. We began with a description of a real-world situation, translated it into mathematical equations, and then employed algebraic techniques to find the solutions. This process underscores the importance of mathematical literacy in everyday life, enabling us to analyze situations, make informed decisions, and solve practical problems. The ability to translate real-world scenarios into mathematical models is a valuable skill that transcends specific disciplines and empowers individuals to navigate complex situations effectively.
Ultimately, our journey into the geometry of Mr. Chaudhary's fish pond serves as a reminder that mathematics is not just an abstract subject, but a powerful tool for understanding and shaping the world around us. From calculating dimensions to determining percentage increases, mathematical principles provide a framework for analyzing spatial relationships, optimizing designs, and solving practical problems. By embracing mathematical thinking, we can unlock new insights and create innovative solutions in diverse fields, from aquaculture to engineering and beyond. The exploration of Mr. Chaudhary's pond has illuminated the beauty and utility of mathematics, inspiring us to further explore its vast potential.
