Quadratic Equation: Pool Dimensions Problem Solved!

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Hey guys! Ever stumbled upon a math problem that looks like a real-world puzzle? That's exactly what we're diving into today. We're going to break down a classic problem involving a rectangular pool, a fence, and a little bit of quadratic equations. Trust me, it's not as scary as it sounds! We'll tackle it together, step by step, and by the end, you'll be a pro at solving these kinds of problems. So, grab your thinking caps, and let's get started!

Understanding the Problem: Pool and Fence

So, let's break down this word problem piece by piece. Our main keyword here is the quadratic equation arising from a geometric scenario. Imagine a rectangular swimming pool. The problem tells us something important: the length of the pool is double its width. This is our first key piece of information. We can start thinking about how to represent this mathematically. Now, picture a fence built around the pool, but not right against the edge. It's one meter away from all sides of the pool. This fence creates a larger rectangle, and we know the area inside this fence: 40 square meters. That's the area our quadratic equation will help us figure out.

Why is this important? Well, these types of problems aren't just abstract math. They show up in real-world situations, from designing gardens to planning construction projects. Understanding how to translate a word problem into an equation is a crucial skill. We need to find the relationship between the pool's dimensions (length and width) and the fenced area. Remember, the fence adds one meter to each side of the pool, so we need to account for that when setting up our equation. And what is a quadratic equation? It’s a polynomial equation of the second degree. A general form is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. The solutions to this equation will give us the possible values for the width of the pool, which we can then use to find the length.

Visualizing the problem can be super helpful. Try drawing a diagram! Sketch a rectangle for the pool, then draw a larger rectangle around it to represent the fence. Label the width of the pool as 'w' and the length as '2w' (since it's double the width). Now, think about how the fence changes those dimensions. The fenced area's width will be 'w + 2' (one meter on each side), and the length will be '2w + 2'. We now have expressions for the dimensions of the fenced area, and we know its area is 40 m². This is where our quadratic equation starts to take shape.

Setting Up the Quadratic Equation

Okay, let's get down to the nitty-gritty of forming our quadratic equation. Remember, the area of a rectangle is simply length times width. We know the fenced area is 40 square meters, and we have expressions for the length and width of the fenced area in terms of 'w' (the pool's width). The key here is translating the word problem into mathematical expressions. We've already established that:

  • Pool width: w
  • Pool length: 2w
  • Fenced area width: w + 2
  • Fenced area length: 2w + 2

Since the area of the fenced rectangle is 40 m², we can write the equation: (w + 2)(2w + 2) = 40. Guys, this is the foundation of our problem! This equation represents the relationship between the pool's width and the fenced area. But it's not in the standard quadratic form yet. We need to expand it and rearrange the terms. Expanding the left side of the equation, we get: 2w² + 2w + 4w + 4 = 40. Now, let's simplify and move all the terms to one side to get our standard quadratic equation form (ax² + bx + c = 0). Combining like terms, we have: 2w² + 6w + 4 = 40. Subtracting 40 from both sides, we arrive at our quadratic equation: 2w² + 6w - 36 = 0. This is it! This equation holds the key to finding the pool's width.

Why is this a quadratic equation? Notice the term '2w²'. The highest power of the variable 'w' is 2, which makes it a quadratic equation. These equations have a unique shape when graphed (a parabola), and they can have up to two solutions. Finding those solutions is our next step. Before we jump into solving, let's think about what our solutions will represent. Since 'w' represents the width of the pool, a solution must be a positive number. A negative width wouldn't make sense in the real world. So, we'll need to keep that in mind when we interpret our answers.

Solving the Quadratic Equation

Alright, now for the exciting part: solving the quadratic equation! We have 2w² + 6w - 36 = 0. There are a few ways we can tackle this. One common method is factoring, but in this case, it might not be the most straightforward. Another powerful tool is the quadratic formula. It's a formula that works for any quadratic equation, no matter how messy it looks. Before we dive into the formula, let's simplify our equation a bit. Notice that all the coefficients (2, 6, and -36) are divisible by 2. Dividing the entire equation by 2 makes it easier to work with: w² + 3w - 18 = 0. Much cleaner, right? Now, let's talk about the quadratic formula. It states that for an equation of the form ax² + bx + c = 0, the solutions for x are given by: x = (-b ± √(b² - 4ac)) / (2a).

Let's identify a, b, and c in our simplified equation (w² + 3w - 18 = 0). Here, a = 1 (the coefficient of w²), b = 3 (the coefficient of w), and c = -18 (the constant term). Now we just plug these values into the quadratic formula! So, we have: w = (-3 ± √(3² - 4 * 1 * -18)) / (2 * 1). Let's simplify this step by step. First, calculate the expression under the square root: 3² - 4 * 1 * -18 = 9 + 72 = 81. The square root of 81 is 9. Now our equation looks like this: w = (-3 ± 9) / 2. This gives us two possible solutions:

  • w = (-3 + 9) / 2 = 6 / 2 = 3
  • w = (-3 - 9) / 2 = -12 / 2 = -6

We have two solutions: w = 3 and w = -6. But remember what 'w' represents – the width of the pool. Can a width be negative? Nope! So, we discard the solution w = -6. This leaves us with w = 3 meters. This is the width of the pool! Finding the solutions to the quadratic equation is crucial, but understanding which solutions make sense in the context of the problem is just as important.

Finding the Pool's Dimensions and Verifying the Solution

We've cracked the code! We found that the width (w) of the pool is 3 meters. But we're not quite done yet. The problem also mentions the length of the pool, which is twice its width. So, the length is 2 * 3 = 6 meters. Now we know the dimensions of the pool: 3 meters wide and 6 meters long. But how do we know if we're right? This is where verification comes in. We need to check if these dimensions, along with the fence, give us the fenced area of 40 square meters that the problem stated.

Remember the fenced area dimensions? They were w + 2 and 2w + 2. Plugging in our value for w (3 meters), we get: Fenced area width: 3 + 2 = 5 meters Fenced area length: 2 * 3 + 2 = 8 meters. Now, let's calculate the fenced area: 5 meters * 8 meters = 40 square meters. Bingo! It matches the information given in the problem. This confirms that our solution is correct. We've successfully used the quadratic equation to find the dimensions of the pool based on the fenced area. Guys, this is how you solve real-world problems using math! It's not just about crunching numbers; it's about understanding the relationships between different quantities and using equations to represent those relationships.

Let's recap the entire process:

  1. We carefully read and understood the word problem, identifying the key information: the relationship between the pool's length and width, the fence around the pool, and the fenced area.
  2. We translated the word problem into mathematical expressions, representing the pool's width as 'w' and expressing the other dimensions in terms of 'w'.
  3. We set up the quadratic equation (w + 2)(2w + 2) = 40, representing the fenced area.
  4. We simplified the equation and rearranged it into the standard quadratic form: 2w² + 6w - 36 = 0.
  5. We solved the quadratic equation using the quadratic formula, obtaining two solutions for 'w'.
  6. We discarded the negative solution (w = -6) because a width cannot be negative.
  7. We identified the valid solution: w = 3 meters (the pool's width).
  8. We calculated the pool's length (6 meters) using the given relationship (length = 2 * width).
  9. We verified our solution by plugging the dimensions into the expressions for the fenced area and confirming that it matched the given area (40 square meters).

Conclusion: Mastering Quadratic Equations in Real-World Scenarios

So, there you have it! We've successfully navigated the pool and fence problem using our understanding of quadratic equations. This wasn't just about memorizing formulas; it was about applying mathematical concepts to a real-world scenario. The ability to translate word problems into equations is a valuable skill that goes beyond the classroom. It's used in engineering, architecture, finance, and many other fields.

The key takeaway is that quadratic equations aren't just abstract mathematical concepts. They're powerful tools for solving practical problems. By breaking down the problem into smaller steps, setting up the equation carefully, and understanding the meaning of the solutions, you can tackle even the most challenging math puzzles. Guys, remember to practice these steps with other similar problems to really solidify your understanding. Keep an eye out for everyday situations where you can apply your newfound quadratic equation skills. You might be surprised where they pop up! Keep practicing, keep exploring, and most importantly, keep having fun with math!