Stacey's Square Cloth Problem Solving For Side Length

In this mathematical exploration, we delve into a fascinating problem involving geometric transformations and area relationships. Our central figure, Stacey, possesses a square piece of cloth, the very foundation of our puzzle. This seemingly simple object undergoes a transformation, a carefully executed reduction in size that unveils a captivating mathematical relationship. The challenge lies in deciphering the original dimensions of this square, a task that demands a blend of algebraic manipulation and geometric intuition. So, let us embark on this journey, armed with the tools of mathematics, to unravel the secrets hidden within Stacey's cloth conundrum.

Setting the Stage: Understanding the Initial Square

At the heart of our problem lies the original square piece of cloth, a geometric figure defined by its four equal sides and four right angles. To embark on our mathematical journey, it's crucial to grasp the fundamental properties of a square, particularly its area. The area of a square, a measure of the space it occupies, is elegantly calculated by squaring the length of one of its sides. This simple yet profound relationship forms the cornerstone of our investigation.

Let's denote the side length of the original square as 's'. This seemingly simple variable becomes our key to unlocking the puzzle. Armed with this notation, we can express the area of the original square as s², a concise mathematical representation of its spatial extent. This initial step, the translation of the geometric object into an algebraic expression, lays the foundation for our subsequent analysis.

Imagine Stacey holding this square of cloth, its dimensions yet unknown. The challenge before us is to determine the value of 's', the side length that defines this square. This task requires us to consider the subsequent transformation, the act of cutting away portions of the cloth, and its impact on the area. By carefully analyzing the relationship between the original square and the smaller square that emerges, we can deduce the elusive value of 's'.

The area of the original square, s², serves as our starting point, the benchmark against which we will compare the area of the smaller square. This comparison, expressed as a mathematical equation, will ultimately lead us to the solution. As we delve deeper into the problem, remember the fundamental connection between the side length and the area of a square, a relationship that guides our every step.

The Transformation: Cutting Away and Creating a Smaller Square

Stacey's square undergoes a meticulous transformation, a reduction in size achieved by cutting away 3 inches from both its length and width. This act of subtraction transforms the original square into a smaller one, a geometric metamorphosis that alters its dimensions and consequently its area. To fully comprehend the impact of this transformation, we must carefully analyze how the side length of the square changes.

If the original square had a side length of 's' inches, then after cutting away 3 inches from both the length and width, the new side length becomes (s - 3) inches. This simple expression captures the essence of the transformation, the reduction in size that stems from Stacey's precise cuts. The smaller square, now defined by this new side length, possesses a different area, a value directly influenced by the amount of cloth removed.

Visualizing this transformation is crucial. Imagine the original square, its sides stretching outwards, and then picture Stacey carefully snipping away 3 inches from each side. The result is a smaller square, nestled within the confines of the original, its dimensions diminished yet its shape preserved. This geometric visualization aids our understanding of the mathematical relationship that governs the transformation.

The area of this smaller square, a key element in our puzzle, can be expressed as (s - 3)². This algebraic representation captures the essence of the transformation, the squaring of the reduced side length. This expression, derived from the geometric manipulation, forms a crucial link in our mathematical chain, connecting the original square to its transformed counterpart.

As we progress towards the solution, remember the impact of Stacey's cuts. The subtraction of 3 inches from each side, a seemingly simple act, has profound consequences for the area of the square. By carefully considering this transformation, we can establish the crucial equation that will unlock the mystery of the original square's side length.

The Area Relationship: Unveiling the Mathematical Connection

The heart of our problem lies in the relationship between the areas of the two squares, the original and the smaller one. The problem states that the area of the smaller square is one-fourth the area of the original square. This seemingly simple statement, expressed as a mathematical equation, forms the cornerstone of our solution. By translating this verbal relationship into algebraic terms, we can unlock the secrets of Stacey's cloth conundrum.

Recall that the area of the original square is s², where 's' represents the original side length. The area of the smaller square, after the transformation, is (s - 3)². Now, let's express the given relationship mathematically: the area of the smaller square is equal to one-fourth the area of the original square. This translates to the following equation:

(s - 3)² = (1/4)s²

This equation, a concise representation of the problem's core information, becomes our key to solving for 's'. It encapsulates the relationship between the original side length and the reduction in area caused by Stacey's cuts. By manipulating this equation, using the tools of algebra, we can isolate 's' and determine its value.

This equation represents a bridge, connecting the geometric world of squares and areas to the algebraic realm of variables and equations. It allows us to translate the problem's narrative into a solvable mathematical form. The power of algebra lies in its ability to express relationships concisely and to provide the tools for solving for unknown quantities.

As we prepare to solve this equation, remember the significance of the area relationship. It's the linchpin of our solution, the connection that allows us to determine the original side length of the square. By carefully applying algebraic techniques, we can unravel the value of 's' and bring closure to Stacey's cloth conundrum.

Solving the Equation: Unlocking the Value of 's'

With our equation firmly established, the next step is to solve for 's', the original side length of the square. This involves a series of algebraic manipulations, each step bringing us closer to the solution. Let's embark on this mathematical journey, carefully applying the rules of algebra to isolate 's' and unveil its value.

Our equation, as a reminder, is:

(s - 3)² = (1/4)s²

First, let's expand the left side of the equation, the squared term (s - 3)². This involves applying the distributive property, also known as the FOIL method, to obtain:

s² - 6s + 9 = (1/4)s²

Now, to simplify the equation, let's eliminate the fraction by multiplying both sides by 4:

4(s² - 6s + 9) = s²

Distributing the 4 on the left side, we get:

4s² - 24s + 36 = s²

Next, let's move all the terms to one side of the equation to set it equal to zero. Subtracting s² from both sides, we obtain:

3s² - 24s + 36 = 0

Now, we have a quadratic equation in the standard form. To simplify further, we can divide all terms by 3:

s² - 8s + 12 = 0

This simplified quadratic equation can be factored. We are looking for two numbers that multiply to 12 and add up to -8. These numbers are -6 and -2. Therefore, we can factor the equation as follows:

(s - 6)(s - 2) = 0

Now, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible solutions:

s - 6 = 0 or s - 2 = 0

Solving for 's' in each case, we get:

s = 6 or s = 2

We have two potential solutions for the original side length, 6 inches and 2 inches. However, we must consider the context of the problem. Stacey cut 3 inches from each side of the square. If the original side length was 2 inches, cutting 3 inches would result in a negative side length, which is not geometrically possible. Therefore, the only valid solution is:

s = 6

Thus, the original side length of Stacey's square piece of cloth was 6 inches.

The Solution: Stacey's Original Square Unveiled

After a journey through geometric transformations and algebraic manipulations, we have arrived at the solution to Stacey's cloth conundrum. The original side length of her square piece of cloth was 6 inches. This seemingly simple answer is the culmination of our efforts, a testament to the power of mathematical reasoning.

We began by defining the problem, understanding the geometric properties of squares and the concept of area. We then translated the problem's narrative into algebraic terms, representing the unknown side length as 's' and expressing the areas of both squares in terms of this variable. The crucial step was establishing the relationship between the areas, the fact that the smaller square's area was one-fourth that of the original.

This relationship led us to a quadratic equation, a mathematical expression that captured the essence of the problem. By carefully applying algebraic techniques, we solved this equation, obtaining two potential solutions for 's'. However, the context of the problem, the act of cutting 3 inches from each side, allowed us to eliminate one solution as geometrically impossible.

The final answer, 6 inches, represents the original dimension of Stacey's square, the foundation upon which our mathematical exploration was built. This solution not only answers the specific question posed but also highlights the power of mathematics in solving real-world problems. By combining geometric intuition with algebraic tools, we were able to unravel the secrets hidden within Stacey's cloth conundrum.

This problem serves as a reminder that mathematics is not just an abstract discipline but a powerful tool for understanding and solving problems in the world around us. From geometric transformations to algebraic equations, mathematics provides a framework for analyzing and interpreting the world, allowing us to make sense of complex relationships and arrive at precise solutions.

Therefore, the side length of the original square was 6 inches.