Calculate Area Of Circular Path Around Garden A Geometry Problem

Hey guys! Ever wondered how to calculate the area of a path surrounding a circular garden? Let's dive into a fun math problem that will show you exactly how to do it. This is super practical for anyone planning a garden or just curious about geometry in the real world. We'll break it down step by step, so don't worry if it sounds intimidating at first. By the end, you'll be a pro at calculating areas of circular paths!

Understanding the Problem: Garden Path Area Calculation

So, here's the scenario: imagine a beautiful circular garden with a radius of 8 feet. Now, picture a path, 3 feet wide, encircling this garden. Our mission, should we choose to accept it (and we do!), is to find the approximate area of just the path. We'll use 3.14 as our trusty approximation for $\pi$. This is a classic problem that combines basic geometry with a bit of real-world application. Before we jump into the calculations, let's make sure we understand the key concepts: radius, area of a circle, and how to deal with areas of composite shapes. Remember, the radius is the distance from the center of the circle to any point on its edge. The area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius. And when we have a shape like our path (which is essentially a larger circle with a smaller circle cut out), we need to think about subtracting areas to find the area of the remaining part. Think of it like cutting a donut out of a bigger piece of dough – we want to know how much dough is left after we remove the donut hole!

When approaching this problem, it’s crucial to visualize the geometry involved. We aren't just dealing with one circle, but two: the inner circle representing the garden and the outer circle encompassing both the garden and the path. The path itself is the region between these two circles. To find the area of this path, we'll need to calculate the area of both circles separately and then subtract the area of the smaller circle (the garden) from the area of the larger circle (garden plus path). This subtraction is key because it isolates the area of the path alone. We'll use the formula $A = \pi r^2$ for both circles, but the $r$ (radius) will be different for each. For the garden, the radius is given as 8 feet. For the larger circle, we need to consider the width of the path, which adds to the garden's radius. This is where careful reading and understanding of the problem statement is essential. So, gear up your math brains, and let's get ready to calculate!

One of the most common pitfalls in problems like these is forgetting to account for the path's width when calculating the radius of the outer circle. It's tempting to just use the garden's radius, but remember, the path extends beyond the garden. This is where a simple diagram can be incredibly helpful. Sketching out the two circles – one for the garden, one for the garden plus the path – can make it much clearer what we need to calculate. Another important point is to keep track of your units. We're dealing with feet for the radii and will end up with square feet for the area. Including units in your calculations can help prevent errors and ensure your final answer is in the correct format. Finally, remember that we're using an approximation for $\pi$ (3.14). This means our final answer will also be an approximation. Don't be surprised if your calculated value isn't exactly one of the multiple-choice options; instead, look for the closest match. Now that we've laid the groundwork and highlighted some key considerations, let's roll up our sleeves and start crunching those numbers!

Step-by-Step Solution: Calculating the Path's Area

Alright, let's break down the solution step by step. First, we need to find the radius of the larger circle, which includes both the garden and the path. The garden has a radius of 8 feet, and the path is 3 feet wide, so the radius of the larger circle is 8 + 3 = 11 feet. Easy peasy, right? Now, we can calculate the area of the larger circle using our formula, $A = \pi r^2$. Plugging in the values, we get $A = 3.14 \times (11)^2 = 3.14 \times 121$. Time for some multiplication! 3. 14 multiplied by 121 equals 379.94 square feet. So, the area of the entire outer circle is approximately 379.94 sq ft.

Next up, we need to calculate the area of the garden itself, which is the smaller circle. We already know the radius of the garden is 8 feet. Using the same formula, $A = \pi r^2$, we get $A = 3.14 \times (8)^2 = 3.14 \times 64$. Again, let's do the math: 3. 14 multiplied by 64 equals 200.96 square feet. So, the area of the garden is approximately 200.96 sq ft. We're getting closer to our final answer! Remember, we want the area of just the path. We've calculated the area of the entire outer circle (garden plus path) and the area of the garden alone. What's the next logical step? You guessed it – we need to subtract the area of the garden from the area of the outer circle. This will leave us with the area of the path.

So, we subtract the area of the garden (200.96 sq ft) from the area of the outer circle (379.94 sq ft): 379.94 - 200.96 = 178.98 square feet. And there you have it! The approximate area of the path alone is 178.98 square feet. We've successfully navigated the geometry and arrived at our solution. It's like we've just cultivated a beautiful mathematical garden! This step-by-step approach is crucial for tackling problems like these. By breaking it down into smaller, manageable calculations, we avoid getting overwhelmed and ensure accuracy. Remember, the key is to understand the underlying concepts, apply the correct formulas, and perform the calculations carefully. And don't forget to double-check your work – a little bit of vigilance can go a long way in avoiding those pesky errors.

Answer and Conclusion: The Path's Area Revealed

Drumroll, please! Looking at our options, the answer that best matches our calculation of 178.98 sq ft is B. 178.98 ft ^2. We nailed it! Feels good, doesn't it? This problem illustrates a fundamental concept in geometry: finding the area of composite shapes by subtracting the areas of simpler shapes. We took a seemingly complex problem and broke it down into manageable steps, applying the formula for the area of a circle and a bit of logical thinking. The result? A clear and accurate answer. This type of problem-solving skill is invaluable, not just in math class, but in many real-world situations.

So, what have we learned today? We've mastered the art of calculating the area of a circular path surrounding a garden (or anything else, for that matter!). We've reinforced our understanding of the formula $A = \pi r^2$ and the importance of careful measurements and calculations. But perhaps more importantly, we've practiced the skill of breaking down a complex problem into smaller, more manageable steps. This is a strategy that can be applied to all sorts of challenges, both inside and outside the world of mathematics. Whether you're planning your own garden path or tackling a tough exam question, remember the power of step-by-step thinking. And always, always double-check your work! Until next time, keep exploring the wonderful world of math, guys! You never know what amazing discoveries you might make.

Original Question

What is the approximate area of the path alone, in square feet, that surrounds a circular garden with a radius of 8 feet, if the path has a width of 3 feet? Use 3.14 for $\pi$.