Hey guys! Today, we're diving into an exciting geometry problem where we'll explore the fascinating world of 3D shapes created by rotating 2D figures. Let's unravel the mystery together!
Problem Statement: The Rotating Rectangle
We have a figure nestled on a coordinate plane, its corners precisely located at the points (2, 0), (2, -2), (6, -2), and (6, 0). Imagine spinning this figure around the x-axis – what kind of 3D shape would magically appear? To ace this challenge, we need to not only identify the shape but also provide a rock-solid explanation and mathematical proof. Let's get started!
Decoding the 2D Figure: A Rectangle Unveiled
Before we jump into the 3D realm, let's take a closer look at our 2D figure. Plotting the points (2, 0), (2, -2), (6, -2), and (6, 0) on a graph reveals a familiar friend: a rectangle. This rectangle is our foundation, the key to unlocking the 3D shape hidden within its rotation. Understanding the dimensions and orientation of this rectangle is crucial for our journey ahead. The sides parallel to the y-axis have a length of 2 units (the difference between 0 and -2), while the sides parallel to the x-axis stretch for 4 units (the difference between 6 and 2). This rectangular blueprint will dictate the form of our 3D creation.
To really grasp what's happening, imagine this rectangle as a door hinged along the x-axis. When we swing this door around, it's going to sweep out a volume, and that volume is the 3D shape we're trying to identify. Think about the path each point of the rectangle traces as it rotates. The points (2,0) and (6,0), which lie directly on the x-axis, will simply stay put. But the other points, (2, -2) and (6, -2), will trace out circles as they spin around. These circular paths are going to be incredibly important in shaping our final 3D object. We need to visualize how these circles, combined with the rectangular shape, come together to form the solid.
Now, consider the distance of the points (2,-2) and (6,-2) from the x-axis. This distance, which is 2 units, will be the radius of the circles they trace out. This is because the x-axis is our axis of rotation, and the distance from any point to the axis of rotation determines the radius of the circle it creates. So, we know we're dealing with circles that have a radius of 2. This knowledge will be crucial when we start to think about the specific dimensions of our 3D shape. By carefully considering these circular paths and the rectangular framework, we can begin to piece together a mental picture of the final form. We're not just blindly guessing at this point; we're using the fundamental principles of geometry and spatial reasoning to build a solid understanding of what's happening.
Visualizing the Rotation: From 2D to 3D
Now comes the exciting part: visualizing the rotation! Imagine taking our rectangle and spinning it a full 360 degrees around the x-axis. What shape is being carved out in space? This is where our spatial reasoning skills come into play. To get a clearer picture, think about the edges of the rectangle as they rotate. The sides parallel to the y-axis will sweep out circular surfaces, while the sides parallel to the x-axis will define the length of our 3D object.
As the rectangle spins, it's like a potter's wheel shaping a clay form. The rotating sides create a hollow space in the middle, and the circular surfaces form the ends of our shape. The distance the rectangle spans along the x-axis (from x = 2 to x = 6) will determine the length of this hollow space. This hollow space is a crucial clue! It suggests we're not dealing with a solid shape, but rather one with a cylindrical void in its center. Think of it like a pipe, or a hollow tube. This is where the concept of a cylinder starts to become important. We know that rotating a rectangle can create cylindrical shapes, but in this case, the hollow center suggests we're dealing with something more specific.
The circles traced by the points (2, -2) and (6, -2) become the circular faces of our 3D shape. These circles have a radius of 2, as we determined earlier. This is because the distance of these points from the x-axis, which is our axis of rotation, is 2 units. So, we now know the radius of the circular ends of our shape. This is another important piece of the puzzle. We're starting to get a good sense of the dimensions of our shape. We know it's going to have circular ends with a radius of 2, and we know it's going to have a hollow cylindrical core. This combination of features is a strong indication of the specific type of 3D shape we're dealing with.
Identifying the 3D Shape: The Cylindrical Shell Emerges
The 3D shape formed by rotating the rectangle around the x-axis is a cylindrical shell (also known as a hollow cylinder or a cylindrical annulus). A cylindrical shell is essentially a cylinder with a smaller cylinder carved out from its center. In our case, the outer cylinder is formed by the rotation of the outer edges of the rectangle, while the inner cylinder is formed by the rotation of the inner space of the rectangle relative to the x-axis.
Think of it like a cardboard tube from a paper towel roll or a toilet paper roll. That's a perfect example of a cylindrical shell. It has an outer cylindrical surface, an inner cylindrical surface, and two circular ends. Our 3D shape, created by rotating the rectangle, will have the same basic form. The dimensions of our cylindrical shell are determined by the dimensions of the rectangle. The height of the cylindrical shell (its length along the x-axis) is the difference between the x-coordinates of the rectangle's vertices, which is 6 - 2 = 4 units. The outer radius of the cylindrical shell is the distance from the x-axis to the farthest edge of the rectangle, which is 2 units (the absolute value of the y-coordinate of the points (2, -2) and (6, -2)). The inner radius of the cylindrical shell is the distance from the x-axis to the closest edge of the rectangle, which in this case is 0 units (since the rectangle sits directly on the x-axis). This means that our inner cylinder has a radius of 0, effectively making it a void.
Proof and Explanation: Why a Cylindrical Shell?
To solidify our answer, let's provide a rigorous explanation and mathematical proof. Here's how we can break it down:
- The Circular Paths: As the rectangle rotates, each point traces a circular path around the x-axis. The radius of each circle is equal to the distance of the point from the x-axis.
- The Outer Cylinder: The points (2, -2) and (6, -2) are 2 units away from the x-axis. Their circular paths form the outer surface of a cylinder with a radius of 2.
- The Inner Cylinder (Void): The space between the rectangle and the x-axis also rotates, creating a cylindrical void in the center. Since the rectangle touches the x-axis, the inner cylinder has a radius of 0, effectively making it a hollow space.
- The Height: The length of the rectangle along the x-axis (from x = 2 to x = 6) determines the height of the cylindrical shell, which is 4 units.
- The Cylindrical Shell: The combination of the outer cylinder and the inner cylindrical void creates a cylindrical shell. This is the fundamental concept behind the shape's formation. It's not just a solid cylinder; it's a cylinder with a hole running through its center. This hollow space is what makes it a cylindrical shell, and it's a direct consequence of the rectangle's position relative to the axis of rotation.
Mathematically, we can think of the volume of the cylindrical shell as the difference between the volume of the outer cylinder and the volume of the inner cylinder. The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. In our case, the outer cylinder has a radius of 2 and a height of 4, so its volume is V_outer = π(2²)(4) = 16π cubic units. The inner cylinder has a radius of 0 and a height of 4, so its volume is V_inner = π(0²)(4) = 0 cubic units. Therefore, the volume of the cylindrical shell is V = V_outer - V_inner = 16π - 0 = 16π cubic units. This calculation further confirms our understanding of the shape and its dimensions.
Conclusion: The Rotating Rectangle's 3D Transformation
So, there you have it! When the figure located at (2, 0), (2, -2), (6, -2), and (6, 0) is rotated around the x-axis, it creates a cylindrical shell. We've not only identified the shape but also provided a clear explanation and mathematical proof to back it up. This exploration beautifully demonstrates how 2D shapes can transform into fascinating 3D objects through rotation. Keep exploring, guys, and happy shape-shifting!
