What Is The Side Measure Of The Cropped Photo In The Equation $(x+2)^2 = 121$, Where The Original Photo's Area Was 121 Square Inches And Was Cropped By 2 Inches On Each Side?

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In this mathematical exploration, we delve into a scenario where Liliana, a meticulous photographer, endeavors to fit a cherished square photo into a frame. The original photograph, boasting an area of 121 square inches, undergoes a transformation as Liliana crops it by 2 inches on each side. Our objective is to decipher the equation $(x+2)^2 = 121$ and unravel the significance of the variable x, which represents the side measure of the cropped photograph.

Understanding the Problem

To embark on this mathematical journey, let's first grasp the essence of the problem. Liliana's original square photo possessed an area of 121 square inches. This implies that each side of the original photograph measured 11 inches (since the square root of 121 is 11). Liliana's decision to crop the photo by 2 inches on each side effectively reduces the dimensions of the photograph, making it smaller to fit snugly into the frame. The equation $(x+2)^2 = 121$ encapsulates this cropping process, where x symbolizes the side measure of the cropped photograph. This equation is the cornerstone of our investigation, providing a mathematical representation of the physical act of cropping the photo. We will dissect this equation, unraveling its meaning and extracting the value of x, which holds the key to the cropped photograph's dimensions. The interplay between the original dimensions, the cropping action, and the resulting cropped size is what makes this problem both intriguing and mathematically relevant. Let's delve deeper into the equation and explore the methods to solve for x, thereby illuminating the dimensions of the cropped photograph.

Dissecting the Equation: Unraveling the Cropped Dimensions

The equation $(x+2)^2 = 121$ serves as a mathematical representation of the cropping process Liliana undertakes. To decipher this equation, we must understand the roles of each component. The variable x represents the unknown side measure of the cropped photograph, which is the crux of our investigation. The addition of 2 to x signifies the original side measure of the photograph before cropping, as Liliana removes 2 inches from each side. The squaring of the expression (x+2) calculates the area of the original photograph, which we know to be 121 square inches. By setting this expression equal to 121, we establish a mathematical relationship that allows us to solve for x. The act of solving for x is akin to reverse-engineering the cropping process, allowing us to determine the dimensions of the cropped photograph based on the original dimensions and the amount cropped. To solve this equation, we can employ several methods. One approach involves taking the square root of both sides of the equation, which simplifies the equation and allows us to isolate x. Alternatively, we can expand the squared expression and rearrange the equation into a quadratic form, which can then be solved using factoring or the quadratic formula. Each method provides a pathway to unveil the value of x, thereby revealing the dimensions of the cropped photograph and resolving Liliana's framing puzzle. The equation acts as a mathematical bridge, linking the original dimensions, the cropping action, and the final cropped size, making it a powerful tool in solving this problem.

Solving for x: Unveiling the Cropped Side Measure

Now, let's embark on the process of solving the equation $(x+2)^2 = 121$ to determine the value of x, which represents the side measure of the cropped photograph. One efficient method involves taking the square root of both sides of the equation. This operation cancels out the square on the left side, simplifying the equation to: $x+2 = ±11$. Note that we consider both positive and negative square roots of 121, as both 11 and -11, when squared, yield 121. This step is crucial in capturing all possible solutions for x. Next, we isolate x by subtracting 2 from both sides of the equation: $x = -2 ± 11$. This yields two potential solutions for x: $x = -2 + 11 = 9$ and $x = -2 - 11 = -13$. However, in the context of this problem, a negative side measure is not physically meaningful. Therefore, we discard the solution x = -13. This leaves us with the solution x = 9. This implies that the side measure of the cropped photograph is 9 inches. We have successfully solved the equation and extracted the value of x, which provides the answer to our central question: what are the dimensions of Liliana's cropped photograph? The process of solving for x is a testament to the power of algebraic manipulation, allowing us to transform an equation into a solution that provides meaningful insights into the physical problem at hand. The positive solution for x confirms our intuition that the cropped photograph should have a smaller side measure than the original, and it aligns perfectly with the context of Liliana's framing endeavor.

Interpreting the Solution: The Cropped Photograph's Dimensions

Having solved the equation and determined that x = 9, we now turn our attention to interpreting the solution in the context of the problem. Recall that x represents the side measure of the cropped photograph. Therefore, our solution implies that each side of the cropped photograph measures 9 inches. This finding is significant because it reveals the dimensions of the photograph after Liliana's cropping. The original photograph, with sides measuring 11 inches, was reduced in size by 2 inches on each side, resulting in the cropped photograph with sides measuring 9 inches. This reduction in size allows the photograph to fit perfectly into the frame Liliana has chosen. The solution x = 9 not only provides a numerical answer but also paints a clear picture of the physical transformation the photograph undergoes. The act of cropping, represented mathematically by the equation $(x+2)^2 = 121$, is now fully understood in terms of its impact on the photograph's dimensions. The cropped photograph, measuring 9 inches on each side, is smaller and more compact, making it suitable for the frame while still preserving the essence of the original image. The interpretation of the solution is a crucial step in the problem-solving process, bridging the gap between mathematical abstraction and real-world application. It allows us to connect the numerical value of x to the physical dimensions of the cropped photograph, providing a complete and meaningful answer to the problem.

Verifying the Solution: Ensuring Accuracy and Consistency

To ensure the accuracy and reliability of our solution, it is essential to verify it within the context of the problem. We have determined that x = 9, which represents the side measure of the cropped photograph. To verify this solution, we can substitute it back into the original equation, $(x+2)^2 = 121$, and check if the equation holds true. Substituting x = 9, we get: $(9+2)^2 = 11^2 = 121$, which is indeed true. This confirms that our solution is consistent with the original equation. Furthermore, we can also verify the solution by considering the physical dimensions of the original and cropped photographs. The original photograph had sides measuring 11 inches, and Liliana cropped 2 inches from each side. This means the cropped photograph should have sides measuring 11 - 2 - 2 = 9 inches, which matches our solution for x. This independent verification reinforces our confidence in the accuracy of the solution. The process of verification is a cornerstone of mathematical rigor, ensuring that our solution not only satisfies the equation but also aligns with the physical constraints of the problem. By verifying our solution, we eliminate the possibility of errors and gain a deeper understanding of the problem and its solution. The consistent results obtained through both substitution and dimensional analysis solidify the validity of our answer, providing a resounding confirmation that the side measure of the cropped photograph is indeed 9 inches.

Conclusion: A Cropped Photo, A Solved Puzzle

In this mathematical journey, we successfully deciphered the dimensions of Liliana's cropped photograph. Starting with the equation $(x+2)^2 = 121$, we embarked on a step-by-step exploration, unraveling the significance of the variable x and its connection to the cropped photograph's side measure. Through algebraic manipulation, we solved for x and obtained the value 9, which represents the side measure of the cropped photograph in inches. We then interpreted this solution in the context of the problem, understanding that the cropped photograph measures 9 inches on each side, a reduction from the original 11 inches due to Liliana's cropping. To ensure the accuracy of our solution, we verified it through substitution and dimensional analysis, confirming its consistency with the original equation and the physical constraints of the problem. This exercise highlights the power of mathematics in modeling real-world scenarios, allowing us to translate a physical action like cropping a photograph into a mathematical equation and then solve it to gain insights into the resulting dimensions. The problem-solving process, from understanding the equation to interpreting the solution, showcases the interconnectedness of mathematical concepts and their relevance to everyday situations. Liliana's photo cropping puzzle serves as a compelling example of how mathematical tools can be used to unravel seemingly complex scenarios, providing clear and concise answers that have practical significance. The cropped photo, now measuring 9 inches on each side, stands as a testament to the successful application of mathematical principles in solving a real-world problem.

Keywords

Liliana, photo cropping, square photo, area, equation, side measure, cropped photograph, $(x+2)^2 = 121$, mathematical problem, solution, dimensions, algebra, verification.