Swimming Pool Dimensions How To Solve For Length And Width
Finding the dimensions of a swimming pool given its area and the relationship between its length and width involves solving a quadratic equation. This article will delve into the step-by-step process of determining the length and width of the pool, providing a clear and comprehensive explanation for those who may be unfamiliar with this type of problem. This exploration will not only provide the solution but also enhance your understanding of applying mathematical concepts in real-world scenarios. Let's embark on this mathematical journey to unravel the dimensions of our swimming pool.
Problem Statement: Unveiling the Pool's Measurements
The problem states that the length of the swimming pool is 8 meters longer than its width, and the total area of the pool is 165 square meters. Our main goal here is to find out the exact length and width of the pool. This is a classic problem that requires us to translate the given information into algebraic equations and then solve them. Understanding how to set up and solve these equations is crucial not only for math problems but also for various real-life scenarios where you need to calculate dimensions or areas.
Setting Up the Equations: Translating Words into Math
To tackle this problem, we first need to convert the word problem into mathematical expressions. Let's denote the width of the pool as 'w' meters. According to the problem, the length of the pool is 8 meters longer than its width, so we can express the length as 'w + 8' meters. The area of a rectangle (which a swimming pool typically is) is calculated by multiplying its length and width. Given that the area of the pool is 165 square meters, we can set up the following equation:
Area = Length × Width 165 = (w + 8) × w
This equation represents the relationship between the width, length, and area of the pool. By solving this equation, we can find the value of 'w,' which will then allow us to determine the length. The next step is to expand and rearrange this equation into a standard quadratic form, which will make it easier to solve.
Expanding and Rearranging: Preparing for the Quadratic Formula
Now that we have our equation, 165 = (w + 8) × w, we need to expand and rearrange it into a more manageable form. This involves distributing the 'w' across the terms in the parentheses:
165 = w² + 8w
To solve this, we need to set the equation to zero, which is the standard form for a quadratic equation (ax² + bx + c = 0). We can achieve this by subtracting 165 from both sides of the equation:
w² + 8w - 165 = 0
This equation is now in the standard quadratic form, where a = 1, b = 8, and c = -165. With the equation in this form, we can use various methods to solve for 'w,' such as factoring, completing the square, or using the quadratic formula. Each method has its advantages, but for this problem, we'll use the quadratic formula, as it is a reliable method for solving any quadratic equation.
Solving the Quadratic Equation: Applying the Formula
The quadratic formula is a powerful tool for solving equations of the form ax² + bx + c = 0. The formula is:
w = (-b ± √(b² - 4ac)) / (2a)
In our equation, w² + 8w - 165 = 0, we have a = 1, b = 8, and c = -165. Plugging these values into the quadratic formula, we get:
w = (-8 ± √(8² - 4 × 1 × (-165))) / (2 × 1)
Let's simplify this step by step. First, calculate the value inside the square root:
√(8² - 4 × 1 × (-165)) = √(64 + 660) = √724
Now, we continue with the formula:
w = (-8 ± √724) / 2
We can approximate √724 as 26.91 (rounded to two decimal places). Now we have:
w = (-8 ± 26.91) / 2
This gives us two possible solutions for 'w':
w₁ = (-8 + 26.91) / 2 = 18.91 / 2 = 9.455 w₂ = (-8 - 26.91) / 2 = -34.91 / 2 = -17.455
Since the width of a pool cannot be a negative value, we discard the negative solution. Therefore, the width of the pool is approximately 9.455 meters. Now that we have the width, we can calculate the length.
Determining the Length: Completing the Dimensions
Now that we have found the width (w) to be approximately 9.455 meters, we can calculate the length using the relationship we established earlier: length = w + 8. Plugging in the value for w, we get:
Length = 9.455 + 8 = 17.455 meters
So, the length of the swimming pool is approximately 17.455 meters. We now have both the width and the length of the pool, which completes the solution to our problem. To summarize, the width is approximately 9.455 meters, and the length is approximately 17.455 meters. It's always a good practice to check our solution to ensure it makes sense in the context of the original problem.
Verification: Ensuring the Solution's Accuracy
To verify our solution, we need to ensure that the calculated dimensions (width ≈ 9.455 meters and length ≈ 17.455 meters) satisfy the conditions given in the problem. The conditions were that the length is 8 meters longer than the width and the area of the pool is 165 square meters. Let's check each condition:
- Length is 8 meters longer than width:
- 17.455 ≈ 9.455 + 8
- 17.455 ≈ 17.455 (This condition is satisfied)
- Area of the pool is 165 square meters:
- Area = Length × Width
- Area ≈ 17.455 × 9.455
- Area ≈ 165.00 (This condition is also satisfied)
Since both conditions are met, our solution is accurate. This verification step is crucial in problem-solving as it confirms that our calculations and logic are correct. Now that we have confidently determined the dimensions of the swimming pool, let's discuss the significance of understanding and solving such problems.
Significance of Solving Quadratic Equation Problems
Solving problems involving quadratic equations, like the one we just tackled, is a fundamental skill in mathematics with far-reaching applications. Understanding how to set up and solve these equations can help in various real-world scenarios, not just in academic settings. Quadratic equations appear in physics, engineering, economics, and many other fields. For instance, they can be used to model projectile motion, calculate areas and volumes, or optimize quantities in business and finance.
Applications in Real Life: Beyond the Classroom
Beyond the specific example of finding the dimensions of a swimming pool, quadratic equations are essential in various real-life applications. Architects and engineers use them to design structures, ensuring they meet certain specifications. Physicists use them to describe the motion of objects under gravity, such as the trajectory of a ball thrown in the air. Economists apply quadratic equations to model supply and demand curves, helping to determine equilibrium prices and quantities. Even in computer graphics, quadratic equations are used to create curves and surfaces.
The ability to solve quadratic equations provides a powerful tool for analyzing and solving problems in diverse fields. This underscores the importance of mastering this concept in mathematics. Furthermore, the problem-solving skills developed through tackling quadratic equation problems, such as translating real-world scenarios into mathematical models, applying appropriate formulas, and verifying solutions, are transferable and valuable in many aspects of life.
Problem-Solving Skills: A Valuable Asset
The process of solving quadratic equation problems enhances critical thinking and analytical skills. It involves several steps, from understanding the problem statement to setting up equations, applying mathematical techniques, and verifying the results. Each step requires careful consideration and logical reasoning.
For example, in our swimming pool problem, we had to translate the given information into algebraic expressions, choose the appropriate formula (the area of a rectangle), solve a quadratic equation, and interpret the results in the context of the problem. This process not only reinforces mathematical concepts but also cultivates problem-solving abilities that are essential in both academic and professional settings. These skills are crucial for making informed decisions, analyzing complex situations, and finding effective solutions to challenges.
Conclusion: Mastering Mathematical Problem-Solving
In conclusion, finding the dimensions of the swimming pool by solving a quadratic equation illustrates the practical application of mathematical concepts in real-world scenarios. We successfully determined the width and length of the pool by translating the given information into an algebraic equation, applying the quadratic formula, and verifying our solution. This exercise highlights the importance of understanding and mastering quadratic equations, as they form a cornerstone of mathematical problem-solving.
Moreover, the problem-solving skills developed through this process are invaluable. The ability to translate real-world problems into mathematical models, apply appropriate techniques, and interpret the results are crucial skills in various fields and aspects of life. By mastering such skills, individuals can approach challenges with confidence and competence, making informed decisions and finding effective solutions. Therefore, the effort invested in understanding and solving mathematical problems, like the one we addressed, is an investment in one's future success and ability to navigate the complexities of the world.
By understanding and solving this problem, we've gained insight not only into the dimensions of a swimming pool but also into the power of mathematics to solve real-world challenges. Whether it's calculating dimensions, modeling physical phenomena, or optimizing quantities, mathematical skills provide a valuable toolset for addressing a wide range of issues. This article serves as a testament to the importance of mathematical literacy and the ability to apply mathematical concepts to practical problems.
