Determining The Value Of Y For A Rectangular Pool

Jun 15, 2025 by ADMIN 50 views

Determining the Value of y for a Rectangular Pool: A Comprehensive Guide

In the realm of geometry and construction, ensuring accuracy in measurements is paramount. Consider a scenario where a contractor is tasked with building a pool, meticulously labeled ABCD on the architectural plans. A critical aspect of this project involves confirming that the pool's shape is indeed a rectangle. To achieve this, we delve into the fundamental properties of rectangles and apply algebraic principles to solve for a specific variable, $y$ . This article aims to provide a comprehensive understanding of the problem, the underlying mathematical concepts, and a step-by-step solution, ensuring clarity and precision in the process.

Understanding the Properties of Rectangles

To accurately determine the value of $y$ that ensures the pool is a rectangle, it's crucial to first understand the defining characteristics of a rectangle. A rectangle is a quadrilateral, a four-sided polygon, with specific properties that distinguish it from other quadrilaterals. These properties include:

Four Right Angles: All four interior angles of a rectangle are right angles, each measuring 90 degrees. This is a fundamental requirement for a quadrilateral to be classified as a rectangle.
Opposite Sides are Parallel and Congruent: The opposite sides of a rectangle are not only parallel but also equal in length. This means that the sides AB and CD are parallel and have the same length, as do sides BC and DA.
Diagonals are Congruent: The diagonals of a rectangle, which are the line segments connecting opposite vertices (corners), are equal in length. In our pool scenario, this means that the diagonal AC must be equal in length to the diagonal BD.

This last property, the congruence of diagonals, is the key to solving our problem. By setting the expressions for the lengths of the diagonals AC and BD equal to each other, we can form an equation and solve for the variable $y$ .

Setting up the Equation

In the given problem, we are provided with the expressions for the lengths of the diagonals AC and BD:

$AC = 15y - 7$
$BD = 2y + 6$

For the pool to be a rectangle, the lengths of these diagonals must be equal. Therefore, we can set up the following equation:

$15y - 7 = 2y + 6$

This equation represents the condition that must be satisfied for the pool to be a rectangle. Now, we proceed to solve this equation for $y$ .

Solving for y: A Step-by-Step Approach

Solving for $y$ involves isolating the variable on one side of the equation. We can achieve this through a series of algebraic manipulations. Here's a step-by-step breakdown:

Combine like terms: Our goal is to group the terms containing $y$ on one side of the equation and the constant terms on the other side. To do this, we can subtract $2y$ from both sides of the equation:

$15y - 7 - 2y = 2y + 6 - 2y$

This simplifies to:

$13y - 7 = 6$
Isolate the term with y: Next, we want to isolate the term containing $y$ , which is $13y$ . To do this, we can add 7 to both sides of the equation:

$13y - 7 + 7 = 6 + 7$

This simplifies to:

$13y = 13$
Solve for y: Finally, to solve for $y$ , we need to divide both sides of the equation by the coefficient of $y$ , which is 13:

$\frac{13y}{13} = \frac{13}{13}$

This simplifies to:

$y = 1$

Therefore, the value of $y$ that ensures the pool is a rectangle is 1.

Verification and Implications

To ensure our solution is correct, we can substitute the value of $y = 1$ back into the original expressions for the lengths of the diagonals:

$AC = 15(1) - 7 = 15 - 7 = 8$
$BD = 2(1) + 6 = 2 + 6 = 8$

Since $AC = BD = 8$ , our solution is verified. This confirms that when $y = 1$ , the diagonals of the pool are equal in length, satisfying the condition for the pool to be a rectangle.

The value of $y$ plays a critical role in the construction of the pool. If $y$ deviates from 1, the diagonals will not be equal, and the pool will not be a perfect rectangle. This could lead to structural issues, aesthetic imperfections, and potentially impact the functionality and safety of the pool. Therefore, precise measurements and adherence to the calculated value of $y$ are essential for the successful construction of a rectangular pool.

Practical Applications and Considerations

This problem highlights the importance of geometry and algebra in real-world applications, particularly in construction and engineering. Understanding the properties of geometric shapes and applying algebraic principles to solve for unknown variables are crucial skills for professionals in these fields. In the context of pool construction, accuracy in measurements and calculations directly impacts the structural integrity and aesthetic appeal of the final product.

Beyond the mathematical solution, there are practical considerations that contractors and builders must take into account. These include:

Material Properties: The materials used in pool construction, such as concrete, fiberglass, or vinyl, have specific properties that can affect the overall dimensions and shape of the pool. Understanding these properties and accounting for them in the design and construction process is essential.
Environmental Factors: Environmental factors, such as soil conditions, weather patterns, and water pressure, can also influence the pool's structure. Proper site preparation and construction techniques are necessary to ensure the pool's longevity and stability.
Building Codes and Regulations: Construction projects must comply with local building codes and regulations. These codes often specify minimum and maximum dimensions, safety requirements, and other factors that can affect the pool's design and construction.

By considering these practical factors in conjunction with the mathematical principles, contractors can ensure the successful and safe construction of a rectangular pool.

Conclusion

In conclusion, determining the value of $y$ that ensures the pool is a rectangle is a practical application of geometric and algebraic principles. By understanding the properties of rectangles, setting up an equation based on the congruence of diagonals, and solving for $y$ , we arrive at the solution $y = 1$ . This value guarantees that the diagonals of the pool are equal in length, satisfying the condition for the pool to be a rectangle. Furthermore, this problem underscores the importance of accuracy in measurements and calculations in construction projects. By combining mathematical precision with practical considerations, contractors can build structures that are not only aesthetically pleasing but also structurally sound and safe. The ability to apply mathematical concepts to real-world scenarios is a valuable skill for professionals in various fields, and this pool construction problem serves as an excellent example of this application.

Therefore, the correct answer is C. 1

This detailed explanation provides a thorough understanding of the problem, the solution, and its implications in a real-world context. It emphasizes the importance of mathematical accuracy and practical considerations in construction projects, making it a valuable resource for students, contractors, and anyone interested in the intersection of mathematics and construction.