Find Alice's Backyard Width: Area & Dimensions

Hey there, math enthusiasts and problem-solvers! Today, we're diving into a classic word problem that combines geometry with a touch of algebra. We're going to help Alice figure out the approximate width of her backyard. Imagine Alice, standing in her yard, wondering about its dimensions. She knows it's a rectangular piece of property, and she's given us two crucial pieces of information: first, that the yard is twice as long as it is wide, and second, that the total area of her yard is 1,000 square meters. Our mission, should we choose to accept it, is to use these clues to find that elusive width. This isn't just about numbers; it's about understanding how shapes and measurements work in the real world, even if it's just a hypothetical backyard. We'll break down the problem step-by-step, making sure to explain each part clearly so you can follow along and even tackle similar problems yourself. Get ready to flex those mathematical muscles!

Unpacking the Problem: Rectangles, Area, and Relationships

Let's start by really getting a handle on what the problem is telling us. We're dealing with a rectangle. Remember, a rectangle has four sides, with opposite sides being equal in length, and all four angles are right angles (90 degrees). The problem states Alice's backyard is rectangular, which is key. The second piece of information is about the relationship between its length and width. It says the backyard is "twice as long as it is wide." This is a crucial ratio we'll need to represent mathematically. If we let 'w' stand for the width of the backyard, then the length ('l') must be '2w' because it's twice the width. So, we have:

• Width = w
• Length = 2w

Now, the third piece of information is the total area of the yard, which is 1,000 square meters ($1,000 m^2$). The area of any rectangle is calculated by multiplying its length by its width. The formula for the area (A) of a rectangle is:

• A = length × width

So, in Alice's case, we can substitute our expressions for length and width into this formula:

• A = (2w) × (w)

And we know that this area 'A' is equal to 1,000 square meters. This gives us our equation:

• 1000 = 2w²

This equation is the mathematical heart of the problem. It combines all the information given into a single, solvable expression. Our goal now is to isolate 'w', the width, to find its approximate value. We'll be using basic algebraic principles to rearrange this equation and solve for 'w'. It's a great example of how mathematical concepts like variables, formulas, and equations can be used to model and solve real-world scenarios, even something as simple as figuring out the size of a yard. The relationship between length and width, along with the area, are the fundamental building blocks for this calculation.

Solving for the Width: Algebra in Action

Now that we have our equation, 1000 = 2w², it's time to put our algebraic skills to the test and solve for 'w', the width of Alice's backyard. Our main objective here is to get 'w' by itself on one side of the equation. We start with the equation:

• 1000 = 2w²

The first step is to isolate the w² term. We can do this by dividing both sides of the equation by 2. This is a valid operation in algebra because whatever you do to one side of an equation, you must do to the other to maintain balance:

• 1000 / 2 = 2w² / 2

This simplifies to:

• 500 = w²

So now we know that the square of the width is 500 square meters. To find the width itself, we need to perform the inverse operation of squaring, which is taking the square root. We will take the square root of both sides of the equation:

• √500 = √w²

This gives us:

• √500 = w

Now, we need to find the approximate value of the square root of 500. This is where we might need a calculator, or we can estimate. The problem asks for the approximate width, which suggests we don't need an exact, infinitely precise decimal. We know that 20² = 400 and 25² = 625. So, the square root of 500 must be somewhere between 20 and 25. Let's try a value in between, say 22. 22² = 484. That's quite close to 500! Let's try 23. 23² = 529. So, the square root of 500 is between 22 and 23, but closer to 22. Using a calculator, the square root of 500 is approximately 22.36.

So, the width 'w' is approximately 22.36 meters. Remember, we are looking for the approximate width, and this value fits our calculation perfectly. This step demonstrates how to manipulate equations to find unknown values, a fundamental skill in mathematics. The process of isolating the variable and applying the inverse operation (square root) is crucial for solving quadratic equations that arise from area problems like this one.

Checking Our Answer and Understanding the Options

We've calculated that the approximate width of Alice's backyard is about 22.36 meters. But the problem provides us with multiple-choice options: A. 22.4 m, B. 44.8 m, C. 15.8 m, and D. 11.23 m. Our calculated value, 22.36 m, is very close to option A, 22.4 m. This gives us confidence that we're on the right track.

Let's take a moment to verify our answer and ensure it makes sense in the context of the original problem. If the width (w) is approximately 22.4 m, then the length (l), which is twice the width, would be approximately 2 * 22.4 m = 44.8 m. Now, let's calculate the area using these dimensions:

• Area = Length × Width
• Area ≈ 44.8 m × 22.4 m

When we multiply these numbers, we get:

• 44.8 × 22.4 = 1003.52 m²

This area, 1003.52 m², is very close to the given area of 1,000 m². The slight difference is due to rounding the width to one decimal place (22.4 m). If we had used our more precise width of 22.36 m, the length would be 44.72 m, and the area would be 22.36 * 44.72 = 999.8752 m², which is even closer to 1,000 m².

Now, let's quickly look at why the other options are incorrect.

• Option B (44.8 m): This value is approximately the length of the backyard, not the width. If the width were 44.8 m, the length would be 89.6 m, leading to a much larger area.
• Option C (15.8 m): If the width were 15.8 m, the length would be 31.6 m. The area would be 15.8 * 31.6 = 499.28 m², which is half of the required area.
• Option D (11.23 m): If the width were 11.23 m, the length would be 22.46 m. The area would be 11.23 * 22.46 = 252.0578 m², which is significantly smaller than 1,000 m².

Therefore, option A, 22.4 m, is the most accurate approximation for the width of Alice's backyard based on the given information and our calculations. It's always a good practice to plug your answer back into the original problem to ensure it holds true. This step not only confirms the correct answer but also reinforces your understanding of the problem's constraints and relationships.

Conclusion: The Width Revealed!

So there you have it! By carefully breaking down the problem, setting up the correct algebraic equation, and solving it step-by-step, we've successfully determined the approximate width of Alice's backyard. We started with the known facts: a rectangular shape, a length twice its width, and a total area of 1,000 m². We translated these facts into the equation 1000 = 2w², where 'w' represents the width. Through algebraic manipulation, we isolated 'w' by dividing by 2 to get w² = 500, and then took the square root to find w ≈ 22.36 m. Comparing this result with the provided options, we confidently identified 22.4 m as the closest approximation.

This exercise is a fantastic reminder of how powerful mathematics is in solving practical problems. Whether you're designing a garden, planning a construction project, or simply trying to understand the space around you, the principles of geometry and algebra are your indispensable tools. The ability to represent relationships with variables and solve equations allows us to quantify the unknown and make informed decisions. We've seen that even a seemingly simple question about a backyard can involve fundamental mathematical concepts. Remember, when faced with a word problem, the key is to read carefully, identify the knowns and unknowns, translate them into mathematical language, and then apply the appropriate techniques to find the solution. And always, always check your answer!

For further exploration into the fascinating world of geometry and area calculations, you can visit Maths is Fun for clear explanations and interactive examples. You might also find the resources at BYJU'S helpful for understanding specific formulas and problem-solving strategies related to rectangles.