Calculating The Volume Of A Hollow Cylindrical Metal Pipe A Step-by-Step Guide

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In the realm of mathematics and engineering, calculating the volume of three-dimensional objects is a fundamental skill. This is particularly true when dealing with hollow cylindrical structures, which are prevalent in various applications, from piping systems to mechanical components. This comprehensive guide will delve into the process of determining the volume of metal required to construct such a hollow cylinder, using a specific example to illustrate the principles involved. We will explore the relevant formulas, provide a step-by-step solution, and offer insights into the underlying concepts. Whether you're a student, engineer, or simply someone with a keen interest in mathematical problem-solving, this article will equip you with the knowledge and skills to confidently tackle similar challenges.

Understanding the Problem: Hollow Cylinder Volume

Before diving into the solution, let's clearly define the problem. We are tasked with finding the volume of metal needed to create a cylindrical pipe with a cylindrical hole cut out of its center. This essentially means we need to calculate the volume of the solid cylindrical shell that remains after the inner cylinder is removed. To achieve this, we'll utilize the concept of subtracting volumes. The volume of the metal will be the difference between the volume of the entire cylinder (outer cylinder) and the volume of the hollowed-out cylinder (inner cylinder). It's crucial to visualize this scenario to grasp the logic behind the calculations. Imagine a solid metal cylinder, and then picture a smaller cylinder being precisely removed from its core. The remaining metal forms the hollow pipe, and its volume is what we aim to determine.

Key Concepts and Formulas for volume calculation

To effectively calculate the volume of a hollow cylinder, it's essential to understand the underlying concepts and formulas. The primary formula we'll use is the formula for the volume of a cylinder: V = πr²h, where:

  • V represents the volume.
  • Ï€ (pi) is a mathematical constant approximately equal to 3.14159.
  • r is the radius of the cylinder's base.
  • h is the height of the cylinder.

In our case, we have two cylinders: the outer cylinder and the inner cylinder. We'll need to apply this formula twice – once for each cylinder – and then subtract the smaller volume from the larger volume. This subtraction represents the removal of the hollowed-out portion, leaving us with the volume of the metal itself. Understanding the relationship between diameter and radius is also crucial. The radius is simply half of the diameter. So, if we're given the diameter, we need to divide it by 2 to obtain the radius for use in the volume formula. These fundamental concepts form the bedrock of our calculation, ensuring we arrive at an accurate solution.

Problem Statement: A Cylindrical Metal Pipe Example

Now, let's introduce the specific problem we'll be solving. We have a cylindrical metal pipe with the following dimensions:

  • Diameter of the outer cylinder: 20 millimeters
  • Height of the cylinder: 21 millimeters
  • Radius of the cylindrical hole (inner cylinder): 6 millimeters

Our objective is to determine the expression(s) that represent the volume of metal needed, in cubic millimeters, to manufacture this pipe. This means we need to set up the volume calculation using the given dimensions and the appropriate formulas. The challenge lies in correctly identifying the radii of both cylinders and applying the subtraction principle to find the metal volume. By carefully substituting the given values into the volume formula and performing the necessary calculations, we can arrive at the solution. This example serves as a practical application of the concepts we discussed earlier, solidifying our understanding of hollow cylinder volume calculations.

Step-by-Step Solution: Calculating the Metal Volume

Let's break down the solution into manageable steps:

  1. Determine the radius of the outer cylinder: The diameter of the outer cylinder is given as 20 millimeters. To find the radius, we divide the diameter by 2: Outer radius (r_outer) = Diameter / 2 = 20 mm / 2 = 10 mm

  2. Calculate the volume of the outer cylinder: Using the formula V = πr²h, we can calculate the volume of the outer cylinder: V_outer = π * (10 mm)² * 21 mm = π * 100 mm² * 21 mm = 2100π mm³

  3. Calculate the volume of the inner cylinder (hole): The radius of the inner cylinder is given as 6 millimeters. Using the same formula, we calculate its volume: V_inner = π * (6 mm)² * 21 mm = π * 36 mm² * 21 mm = 756π mm³

  4. Calculate the volume of the metal: To find the volume of the metal, we subtract the volume of the inner cylinder from the volume of the outer cylinder: V_metal = V_outer - V_inner = 2100π mm³ - 756π mm³ = 1344π mm³

Therefore, the expression representing the volume of metal needed to make the pipe is 1344Ï€ cubic millimeters. This step-by-step approach ensures clarity and accuracy in the calculation process, making it easier to understand and replicate for similar problems.

Expressing the Volume: Different Representations

While 1344π cubic millimeters is the exact volume, it can be expressed in different forms depending on the context and desired level of precision. We can leave the answer in terms of π, as we have done, which provides the most accurate representation. Alternatively, we can approximate π as 3.14159 and perform the multiplication to get a numerical value. This would give us an approximate volume in cubic millimeters. For practical applications, rounding the numerical value to a suitable number of decimal places might be necessary. For example, multiplying 1344 by 3.14159 gives us approximately 4221.15 cubic millimeters. This value could be rounded to 4221 cubic millimeters or 4221.2 cubic millimeters, depending on the required precision. Understanding the different ways to express the volume allows for flexibility and adaptability in various scenarios.

Alternative Expressions and Their Significance

Besides the numerical approximation, there are other ways to express the volume that can be insightful. For example, we can factor out common terms in the original volume subtraction (2100π - 756π) to reveal relationships between the dimensions. We can also express the volume as a product of π and a simplified numerical value, as we did in the final answer (1344π). These alternative expressions can highlight the contribution of each dimension (radius and height) to the overall volume. They can also be useful for comparing the volumes of different hollow cylinders or for optimizing the design of such structures. The ability to manipulate and interpret these expressions is a valuable skill in mathematics and engineering, providing a deeper understanding of the underlying principles.

Importance of Units: Cubic Millimeters (mm³)

In any scientific or engineering calculation, the importance of units cannot be overstated. In our problem, the final answer is expressed in cubic millimeters (mm³). This unit signifies that we are measuring volume, which is a three-dimensional quantity. The