Hollow Cylinder Volume: A Metal Pipe Calculation
Hey guys! Ever wondered how to calculate the volume of metal needed to make a hollow pipe? It's a pretty cool application of geometry, and in this article, we're going to break it down step-by-step. We'll tackle a specific problem: a cylindrical metal pipe with a diameter of 20 millimeters and a height of 21 millimeters, which has a cylindrical hole cut out of its center with a radius of 6 millimeters. Our mission is to find out the volume of metal needed to make this pipe. So, let's dive into the math and see how it's done!
Understanding the Problem: Visualizing the Hollow Cylinder
Before we jump into calculations, let's get a clear picture of what we're dealing with. Imagine a solid cylinder – that's our starting point. This solid cylinder has a diameter of 20 millimeters, meaning its radius (the distance from the center to the edge) is half of that, which is 10 millimeters. It also has a height of 21 millimeters. Now, picture a smaller cylinder being carved out from the very center of this solid cylinder. This smaller, hollowed-out cylinder has a radius of 6 millimeters. What we're left with is a cylindrical pipe, a hollow cylinder, where the metal forms the outer wall.
In essence, the volume of the metal in this pipe is the difference between the volume of the larger, original cylinder and the volume of the smaller, hollowed-out cylinder. To find this, we'll use the formula for the volume of a cylinder: V = πr²h, where V is the volume, r is the radius, and h is the height. Remembering this fundamental formula is key to solving this problem. We'll apply this formula twice – once for the outer cylinder and once for the inner cylinder – and then subtract the volumes to find our answer. Think of it like this: we're finding the volume of the whole and subtracting the volume of the hole to get the volume of the material. This principle of subtracting volumes is a common technique in geometry, especially when dealing with complex shapes formed by removing portions from simpler ones. Getting a good grasp of this concept will definitely help you tackle similar problems in the future. So, keep that image of the solid cylinder with the hole in mind as we move on to the calculations!
Calculating the Volume: Step-by-Step Breakdown
Now that we've visualized the problem, let's get down to the nitty-gritty and calculate the volume of metal needed. Remember, the key idea here is to find the volume of the outer cylinder, then find the volume of the inner cylinder (the hollow part), and finally, subtract the inner volume from the outer volume. This will give us the volume of the metal itself. So, let's grab our calculators and get started!
First, let's tackle the outer cylinder. We know its radius is 10 millimeters and its height is 21 millimeters. Using the formula V = πr²h, we can plug in these values: V_outer = π * (10 mm)² * (21 mm) = π * 100 mm² * 21 mm = 2100π mm³. So, the volume of the outer cylinder is 2100π cubic millimeters. Now, let's move on to the inner cylinder, the hollow part. Its radius is 6 millimeters, and the height is the same as the outer cylinder, 21 millimeters. Applying the same formula, we get: V_inner = π * (6 mm)² * (21 mm) = π * 36 mm² * 21 mm = 756π mm³. This means the volume of the hollow part is 756π cubic millimeters. Almost there! To find the volume of the metal, we simply subtract the volume of the inner cylinder from the volume of the outer cylinder: V_metal = V_outer - V_inner = 2100π mm³ - 756π mm³. This simplifies to V_metal = 1344π mm³. And there you have it! The volume of metal needed to make the pipe is 1344π cubic millimeters. This result can be expressed in terms of π, which is often preferred for precision, or you can use a calculator to approximate it to a decimal value (approximately 4222.3 mm³). Understanding each step, from identifying the radii to applying the formula and subtracting the volumes, is crucial for mastering these types of problems.
Expressing the Volume: Different Mathematical Representations
Okay, we've calculated the volume of the metal in the pipe, which is 1344π cubic millimeters. But here's the thing about math problems – there's often more than one way to express the answer! Understanding these different representations is super important because it shows a deeper understanding of the concepts involved. In this case, we can look at how this volume can be expressed in different mathematical forms, which might be what a test question or a real-world application asks for.
Our current answer, 1344π mm³, is a perfectly valid way to represent the volume. It's an exact value because it keeps π as a symbol, avoiding any rounding errors that might occur if we used an approximation like 3.14. However, sometimes you might need to see this expression broken down further. For instance, you might be asked to show the calculation steps explicitly. In that case, the expression would look like this: π * (10 mm)² * (21 mm) - π * (6 mm)² * (21 mm). This shows clearly that we're subtracting the volume of the inner cylinder from the volume of the outer cylinder. Another way to represent the volume is by factoring out the common terms. Notice that both terms in the expression above have π and 21 mm as factors. We can factor these out to get: π * 21 mm * ((10 mm)² - (6 mm)²). This factored form is mathematically equivalent to our original answer, but it highlights the common factors and can sometimes be easier to work with in further calculations. Simplifying the squares inside the parentheses, we get: π * 21 mm * (100 mm² - 36 mm²) = π * 21 mm * (64 mm²). Multiplying 21 and 64 gives us 1344, so we're back to our original answer of 1344π mm³. The key takeaway here is that there are multiple ways to represent the same volume, and being comfortable with manipulating these expressions is a valuable skill in mathematics and problem-solving. Whether it's the exact form, the expanded form showing the subtraction, or the factored form highlighting common terms, each representation offers a different perspective on the same underlying value. So, keep practicing with these different forms, and you'll become a pro at expressing mathematical concepts in various ways!
Real-World Applications: Where This Math Matters
Alright, we've crunched the numbers and found the volume of metal needed for our cylindrical pipe. But you might be thinking,
