Cuando El Lado De Un Cuadrado Se Incrementa En 30%, El Área Aumenta En 621 M2. ¿Cómo Calcular El Lado Inicial Del Cuadrado?

Introduction: Understanding the Relationship Between Side Length and Area

In the realm of geometry, understanding the relationship between the dimensions of a shape and its area is fundamental. When we talk about squares, this relationship is particularly straightforward yet powerful. A square, defined by its four equal sides and four right angles, has an area that is simply the square of its side length. This means if you know the length of one side, you can easily calculate the area by multiplying that length by itself. However, things get interesting when we start changing the side length and observing how the area responds. This article delves into a specific problem where the side of a square is increased, leading to an increase in its area, and our task is to find the original side length.

The problem at hand presents a scenario where the side of a square is increased by 30%, and as a result, the area increases by 621 square meters. This kind of problem is not just a mathematical exercise; it mirrors real-world scenarios where scaling dimensions can have significant impacts on the area or volume of a space. For instance, architects and engineers frequently deal with such calculations when designing spaces or structures. Understanding how a percentage increase in one dimension affects the overall area is crucial in these fields. Therefore, grasping the concepts involved in this problem provides valuable insights that extend beyond the classroom.

To solve this problem, we will employ algebraic techniques, setting up an equation that relates the original side length to the new side length and their corresponding areas. By carefully working through the equation, we can isolate the variable representing the original side length and find its value. This process not only gives us the answer to the specific problem but also enhances our problem-solving skills in general. The ability to translate a word problem into a mathematical equation and then solve it is a critical skill in mathematics and many other disciplines. So, let's dive into the problem and unravel the mystery of the square's original side length.

Setting Up the Equation: Translating the Word Problem into Algebra

To effectively solve mathematical problems, particularly those presented in a word format, the initial and crucial step involves translating the given information into an algebraic equation. This process requires careful reading and understanding of the problem statement, identifying the knowns and unknowns, and then establishing a relationship between them. In our case, we are dealing with a square whose side length is increased, resulting in a change in its area. Our goal is to find the original side length of the square.

Let's denote the original side length of the square as 'x' meters. Since the area of a square is the side length squared, the original area of the square is x² square meters. The problem states that the side length is increased by 30%. To represent this increase mathematically, we need to calculate 30% of 'x' and add it to the original side length. 30% of x is (30/100) * x, which simplifies to 0.3x. Therefore, the new side length is x + 0.3x, which equals 1.3x meters. The new area of the square, after the side length is increased, is (1.3x)² square meters.

The problem also tells us that the area increases by 621 square meters. This means the new area is the original area plus 621 square meters. We can express this as an equation: (1.3x)² = x² + 621. This equation is the cornerstone of our solution. It mathematically represents the relationship between the original side length, the increased side length, and the change in area. Solving this equation will give us the value of 'x', which is the original side length of the square. This step of setting up the equation is critical because it transforms a word problem into a solvable mathematical form, allowing us to use algebraic techniques to find the solution.

Solving the Equation: Finding the Value of the Original Side

Now that we have successfully translated the word problem into an algebraic equation, the next step is to solve the equation and find the value of the unknown variable, which in our case is the original side length of the square, denoted as 'x'. Our equation is (1.3x)² = x² + 621. To solve this, we need to simplify and isolate 'x' on one side of the equation.

First, let's expand the left side of the equation: (1.3x)² equals 1.69x². So, our equation becomes 1.69x² = x² + 621. Next, we want to get all the terms involving x² on one side of the equation. We can do this by subtracting x² from both sides, which gives us 1.69x² - x² = 621. Simplifying the left side, we have 0.69x² = 621. Now, to isolate x², we need to divide both sides of the equation by 0.69, resulting in x² = 621 / 0.69. Performing this division, we find that x² = 900.

To find the value of 'x', we need to take the square root of both sides of the equation. The square root of x² is 'x', and the square root of 900 is 30. Therefore, x = 30. Since 'x' represents the original side length of the square, we have found that the original side length is 30 meters. It's important to note that when taking the square root, we typically consider both positive and negative solutions. However, in this context, since we are dealing with a physical dimension (the side length of a square), only the positive solution makes sense. This step-by-step solution demonstrates how algebraic manipulation can be used to solve real-world problems, providing a clear and concise method for finding the unknown value.

Verification: Ensuring the Solution Aligns with the Problem Conditions

After arriving at a solution in any mathematical problem, particularly in word problems, it is crucial to verify the solution. This step ensures that the answer we have obtained not only satisfies the equation we set up but also makes logical sense within the context of the original problem. Verification helps to catch any potential errors in our calculations or reasoning and gives us confidence in the correctness of our answer. In our case, we found that the original side length of the square is 30 meters. To verify this, we need to check if this value, when used according to the problem conditions, results in the area increase described.

Let's start with the original square. If the side length is 30 meters, the original area is 30² = 900 square meters. Now, the side length is increased by 30%, which means it is increased by 0.3 * 30 = 9 meters. The new side length is therefore 30 + 9 = 39 meters. The new area of the square is 39² = 1521 square meters. The increase in area is the new area minus the original area, which is 1521 - 900 = 621 square meters. This matches the information given in the problem, which stated that the area increases by 621 square meters when the side length is increased by 30%.

Since our calculated increase in area matches the problem statement, we can confidently say that our solution is correct. This verification process not only confirms the numerical answer but also reinforces our understanding of the relationships between the side length and area of a square. It highlights the importance of not just solving for an answer but also ensuring that the answer is logically consistent with the problem's conditions. This practice is essential for developing strong problem-solving skills and ensuring accuracy in mathematical applications.

Conclusion: Reflecting on the Problem-Solving Process and Implications

In conclusion, we have successfully solved the problem of finding the original side length of a square, given that increasing its side by 30% results in an area increase of 621 square meters. Our journey through this problem has highlighted the importance of several key mathematical concepts and problem-solving strategies. We began by translating the word problem into an algebraic equation, a critical step in mathematical problem-solving. This involved identifying the unknowns, assigning variables, and expressing the relationships between the given quantities in a mathematical form. The equation we derived, (1.3x)² = x² + 621, encapsulated the core information of the problem and set the stage for our solution.

Next, we systematically solved the equation, employing algebraic techniques such as expanding terms, simplifying expressions, and isolating the variable. This process demonstrated the power of algebra in solving geometric problems and underscored the importance of precision in mathematical manipulations. Our solution, x = 30 meters, represented the original side length of the square. However, our work did not end there. We then verified our solution, a crucial step in ensuring the accuracy and validity of our answer. By calculating the new area based on the increased side length and confirming that the area increase matched the problem statement, we gained confidence in our solution.

This problem also illustrates the practical implications of mathematical concepts in real-world scenarios. Understanding how changes in dimensions affect area is essential in various fields, including architecture, engineering, and design. For instance, when scaling a blueprint or designing a structure, it is vital to accurately calculate how changes in length impact the overall area or volume. Moreover, the problem-solving skills honed through this exercise, such as translating word problems into equations and systematically solving them, are transferable to a wide range of situations beyond mathematics. The ability to approach problems logically, break them down into manageable steps, and verify solutions is a valuable asset in any field. Therefore, this problem serves not only as a mathematical exercise but also as a lesson in critical thinking and problem-solving.