Sidewalk Width Around Square Garden: A Math Problem

Let's dive into a fun math problem about calculating the width of a sidewalk around a square garden! This is a classic geometry problem that combines area calculations with a bit of algebra. So, grab your thinking caps, guys, and let's get started!

Problem Statement: Idris's Garden Sidewalk

Our problem: Idris plans to build a sidewalk around a 10-yard-long square garden. The garden and sidewalk together must cover a total area of 400 square yards. We know the garden is a perfect square, so each side of the garden plus the sidewalk will measure ($10 + 2w$) yards, where w represents the width of the sidewalk. Our mission is to find the value of w – the width of that sidewalk.

This problem is a fantastic example of how math concepts can be applied to real-world scenarios. We'll be using our knowledge of area, squares, and a little bit of algebra to solve it. Don't worry, it's not as intimidating as it might sound! We'll break it down step by step, making sure everyone can follow along. Whether you're a math whiz or just brushing up on your skills, this problem offers a great opportunity to sharpen your problem-solving abilities. So, let's roll up our sleeves and get into the solution!

Breaking Down the Problem: Visualizing the Garden and Sidewalk

To really understand this problem, let's visualize what's happening. Imagine a square garden sitting snugly inside a larger square, which includes both the garden and the sidewalk surrounding it. The garden itself is 10 yards on each side. Now, picture a sidewalk of uniform width, w, wrapping around all four sides of the garden. This sidewalk increases the overall dimensions, and that's where the expression ($10 + 2w$) comes in. Think about it: the sidewalk adds w yards on one side and another w yards on the opposite side, hence the "2w".

Key Dimensions and Areas

• Garden: A square with sides of 10 yards.
• Sidewalk: Uniform width w around the garden.
• Combined Area (Garden + Sidewalk): 400 square yards.
• Side Length of Combined Area: ($10 + 2w$) yards.

By visualizing this setup, we can see how the width of the sidewalk directly impacts the total area. The larger the sidewalk width (w), the greater the overall area covered by the garden and the sidewalk. This visualization is crucial because it helps us translate the word problem into mathematical terms. We can now start thinking about how to express the areas mathematically and set up an equation to solve for w. Remember, guys, drawing a diagram can be super helpful in these types of problems – it's a great way to organize your thoughts and see the relationships between different parts of the problem.

Setting Up the Equation: Area Calculations

Now that we have a good visual understanding, let's translate this into math! The key here is to think about areas. We know the area of a square is simply the side length multiplied by itself (side * side, or side²). So, we have two squares to consider:

1. The garden: This is a square with sides of 10 yards. Its area is straightforward: 10 yards * 10 yards = 100 square yards.
2. The garden plus the sidewalk: This is also a square, but its sides are ($10 + 2w$) yards long. Therefore, its area is ($10 + 2w$) * ($10 + 2w$), or ($10 + 2w$)². We also know this combined area is 400 square yards.

Formulating the Equation

Here's where we connect the pieces. The area of the garden plus the sidewalk is 400 square yards. We've already expressed this area as ($10 + 2w$)². So, we can set up the following equation:

($10 + 2w$)² = 400

This equation is the heart of our problem. It mathematically represents the relationship between the sidewalk width (w) and the total area. To solve for w, we need to expand this equation and then use our algebra skills to isolate w. This might seem a bit daunting at first, but don't worry, we'll take it one step at a time. Remember, the goal is to get w by itself on one side of the equation, which will tell us the width of the sidewalk.

Solving the Equation: Finding the Sidewalk Width

Alright, let's tackle that equation: ($10 + 2w$)² = 400. The first step is to expand the squared term. Remember that ($10 + 2w$)² means ($10 + 2w$) multiplied by itself. We can use the FOIL method (First, Outer, Inner, Last) or the distributive property to expand this:

($10 + 2w$)($10 + 2w$) = 10 * 10 + 10 * 2w + 2w * 10 + 2w * 2w = 100 + 20w + 20w + 4w² = 4w² + 40w + 100

Simplified Equation

Now our equation looks like this:

4w² + 40w + 100 = 400

To make it easier to solve, let's subtract 400 from both sides to set the equation to zero:

4w² + 40w + 100 - 400 = 0

4w² + 40w - 300 = 0

Simplifying Further

Notice that all the coefficients are divisible by 4. Dividing the entire equation by 4 simplifies things nicely:

w² + 10w - 75 = 0

Factoring the Quadratic

Now we have a quadratic equation in the standard form (ax² + bx + c = 0). To solve for w, we can try factoring. We need to find two numbers that multiply to -75 and add up to 10. Those numbers are 15 and -5.

So, we can factor the equation as:

(w + 15)(w - 5) = 0

Finding the Solutions

This gives us two possible solutions for w:

• w + 15 = 0 => w = -15
• w - 5 = 0 => w = 5

Interpreting the Solution: Choosing the Right Answer

We've arrived at two possible solutions for w: -15 and 5. But hold on a second! In the real world, the width of a sidewalk can't be a negative number. A negative width doesn't make sense in our context. So, we can discard the solution w = -15.

The Correct Answer

This leaves us with w = 5. This means the width of the sidewalk is 5 yards. This makes perfect sense in the context of our problem. A sidewalk with a width of 5 yards around a 10-yard square garden would indeed increase the total area to 400 square yards.

Verification

Let's quickly verify our answer. If the sidewalk is 5 yards wide, then the side length of the garden plus the sidewalk is 10 yards + 2 * 5 yards = 20 yards. The total area would then be 20 yards * 20 yards = 400 square yards, which matches the given information. Hooray, we got it right!

Conclusion: Math in Action

So, there you have it! We've successfully calculated the width of the sidewalk around Idris's garden. The width, w, is 5 yards. This problem beautifully illustrates how math concepts like area, squares, and algebraic equations can be used to solve practical problems. By visualizing the situation, breaking it down into smaller parts, and applying the right formulas, we were able to find the solution.

Key Takeaways

• Visualization is Key: Drawing a diagram or visualizing the problem can make it much easier to understand.
• Break it Down: Complex problems can be solved by breaking them down into smaller, manageable steps.
• Real-World Applications: Math isn't just about numbers and formulas; it's a powerful tool for solving real-world problems.

I hope you guys enjoyed this mathematical journey! Remember, practice makes perfect, so keep exploring and tackling new challenges. Whether it's calculating sidewalk widths or figuring out other geometric puzzles, math is all around us, waiting to be discovered.