UNIOESTE 2012 Unveiling Dimensions Of A Rectangular Backyard

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In the realm of mathematical challenges, the UNIOESTE 2012 problem presents a fascinating puzzle involving a rectangular backyard. This backyard possesses a unique characteristic: one of its sides is precisely three times the length of the other. Adding another layer of intrigue, the problem states that the perimeter of the backyard, measured in meters, is numerically equal to its area, measured in square meters. Our quest is to determine the length of the longer side of this intriguing backyard.

Deciphering the Problem: A Step-by-Step Approach

To embark on this mathematical journey, we must first translate the problem's description into a language of mathematical expressions. Let us denote the shorter side of the rectangle as 'x' meters. Consequently, the longer side, being three times the length of the shorter side, will be '3x' meters. With these variables in hand, we can now express the perimeter and area of the rectangle in terms of 'x'.

The perimeter of a rectangle is calculated by adding up the lengths of all its sides. In this case, the perimeter would be x + 3x + x + 3x, which simplifies to 8x meters. The area of a rectangle, on the other hand, is found by multiplying its length and width. Therefore, the area of our rectangular backyard is x * 3x, which equals 3x² square meters.

The heart of the problem lies in the statement that the perimeter and area are numerically equal. This allows us to set up an equation: 8x = 3x². This equation serves as the key to unlocking the dimensions of our backyard.

Solving the Equation: Unveiling the Value of 'x'

To solve the equation 8x = 3x², we need to manipulate it algebraically to isolate the variable 'x'. First, we can rearrange the equation to bring all terms to one side: 3x² - 8x = 0. Now, we can factor out an 'x' from both terms on the left side: x(3x - 8) = 0.

This equation reveals two possible solutions for 'x': either x = 0 or 3x - 8 = 0. The solution x = 0 is not physically meaningful in the context of our backyard, as it would imply a rectangle with no dimensions. Therefore, we focus on the second solution, 3x - 8 = 0.

Solving for 'x' in this equation, we add 8 to both sides and then divide by 3, giving us x = 8/3 meters. This value represents the length of the shorter side of our rectangular backyard.

Determining the Longer Side: The Final Piece of the Puzzle

Now that we know the length of the shorter side, we can easily determine the length of the longer side. Recall that the longer side is three times the length of the shorter side. Therefore, the longer side is 3 * (8/3) = 8 meters.

Thus, we have successfully deciphered the dimensions of the rectangular backyard. The shorter side measures 8/3 meters, while the longer side stretches to 8 meters.

Embracing Mathematical Problem-Solving: A Journey of Discovery

The UNIOESTE 2012 problem exemplifies the power and elegance of mathematical problem-solving. By carefully translating the problem's description into mathematical expressions, setting up an equation, and solving for the unknown, we were able to unravel the dimensions of the unique rectangular backyard.

This problem-solving journey underscores the importance of:

  • Understanding the Problem: Grasping the core concepts and relationships presented in the problem statement.
  • Translating into Mathematics: Expressing the problem's information using mathematical symbols, variables, and equations.
  • Strategic Problem-Solving: Employing appropriate mathematical techniques to manipulate equations and isolate unknowns.
  • Interpreting Results: Ensuring that the solutions obtained are physically meaningful and aligned with the problem's context.

Exploring Further: Variations and Extensions

To deepen our understanding and expand our mathematical horizons, we can explore variations and extensions of the UNIOESTE 2012 problem. For instance, we could investigate:

  • Different Ratios: What if the longer side was twice or four times the length of the shorter side?
  • Varying Relationships: What if the perimeter was twice or half the area?
  • Three-Dimensional Extensions: Can we extend this problem to three-dimensional shapes like rectangular prisms?

By exploring these variations, we can further hone our problem-solving skills and gain a deeper appreciation for the beauty and versatility of mathematics.

Concluding Thoughts: The Allure of Mathematical Challenges

The UNIOESTE 2012 problem serves as a testament to the captivating nature of mathematical challenges. These puzzles not only test our analytical abilities but also ignite our curiosity and drive us to seek solutions. By embracing problem-solving as a journey of discovery, we unlock the power of mathematics to illuminate the world around us.

In conclusion, the longer side of the rectangular backyard measures 8 meters, a testament to the power of mathematical deduction and problem-solving.

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Embracing the Human Element in Mathematical Writing

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Conclusion: The Power of Mathematical Storytelling

In conclusion, the UNIOESTE 2012 problem serves as a reminder that mathematics is not just about numbers and equations; it's about storytelling. By crafting compelling narratives around mathematical concepts, we can unlock their power to engage, inspire, and transform the way we see the world. The longer side of the rectangular backyard, a seemingly simple answer, becomes a symbol of the intricate beauty and boundless possibilities that lie within the realm of mathematics.

Therefore, the answer is (b) 8 m.