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Chapter 3: Problem 43

Estimate the volume of material in a cylindrical shell with length \(30 \mathrm{cm},\) radius \(6 \mathrm{cm},\) and shell thickness \(0.5 \mathrm{cm}\).

Short Answer

Expert verified
The volume of material in the shell is approximately 578.15 cm鲁.

Step by step solution

01

Understanding the Cylindrical Shell

The problem involves a cylindrical shell, which means it's a hollow cylinder. To estimate the volume of material in such a shell, find the volume of the larger outer cylinder and subtract the volume of the smaller inner cylinder.
02

Calculate Volume of Outer Cylinder

The outer radius of the cylinder is the sum of the radius of the shell and thickness of the shell: \[ R = r + t = 6 \, \text{cm} + 0.5 \, \text{cm} = 6.5 \, \text{cm} \]The formula for the volume of a cylinder is given by \[ V = \pi R^2 h \]where \( R = 6.5 \, \text{cm} \) and \( h = 30 \, \text{cm} \). So, the volume of the outer cylinder is \[ V_{\text{outer}} = \pi (6.5)^2 \times 30 \approx 3971.07 \, \text{cm}^3 \]
03

Calculate Volume of Inner Cylinder

The radius of the inner cylinder is equal to the original radius of the shell:\[ r = 6 \, \text{cm} \]Using the formula for the volume of a cylinder, the volume of the inner cylinder is\[ V_{\text{inner}} = \pi r^2 h = \pi (6)^2 \times 30 \approx 3392.92 \, \text{cm}^3 \]
04

Find the Volume of the Shell

The volume of the material in the shell is the difference between the volumes of the outer and inner cylinders:\[ V_{\text{shell}} = V_{\text{outer}} - V_{\text{inner}} \]Substituting the values we calculated:\[ V_{\text{shell}} \approx 3971.07 \, \text{cm}^3 - 3392.92 \, \text{cm}^3 = 578.15 \, \text{cm}^3 \]
05

Conclusion

The volume of material in the cylindrical shell is approximately 578.15 \( \text{cm}^3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

The Basics of Geometry in Cylindrical Shells
Geometry is the study of shapes, sizes, and the properties of spaces. It allows us to understand and calculate different dimensions and volumes of various shapes, like cylinders. In a cylindrical shell, the interest lies in both the outer and inner surfaces that make up the hollow part.
These surfaces create a three-dimensional object, which is essentially described by the radius, height, and thickness. Understanding geometry gives us the tools to handle these measurements correctly, providing the foundation for calculating their corresponding volumes.
Here, the cylindrical shell is a specific form of geometry where we observe two main circles (the base and top) and the height, creating a recognizably hollow cylinder.
Understanding Cylinder Volume
The volume of a cylinder is crucial in understanding how much space a shape occupies. A basic cylindrical volume is calculated by using the formula:
  • \( V = \pi R^2 h \) where:
  • \( R \) is the radius,
  • \( h \) is the height,
  • and \( \pi \) is approximately 3.14159.
For a cylindrical shell, you calculate the volumes of both the larger, outer cylinder and the smaller, inner cylinder. You subtract the latter from the former to find the shell volume. This method ensures an accurate measure of just the material making up the walls of the shell, showing the clever application of volume understanding in practical scenarios.
Examining Geometric Shapes: Cylindrical Shells
Cylindrical shells are fascinating geometric shapes. They are examples of bounded, three-dimensional forms which highlight both the inner and outer surfaces. These shapes demonstrate how geometry and mathematical formulas work together to solve real-world problems by calculating everything from construction material needs to fluid capacities.
In academic settings, examining cylindrical shells provides students with examples of where their knowledge of mathematics and geometry has tangible applications. Analyzing the interaction between its radius, height, and thickness makes understanding geometric relationships more comprehensive.
Furthermore, these shapes offer insight into how seemingly simple forms can consist of complex surfaces interacting in space, showcasing the beauty and usability of geometric shapes in both learning and daily life.

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Most popular questions from this chapter

Graph \(y=-2 x \sin \left(x^{2}\right)\) for \(-2 \leq\) \(x \leq 3 .\) Then, on the same screen, graph $$y=\frac{\cos \left((x+h)^{2}\right)-\cos \left(x^{2}\right)}{h}$$ for \(h=1.0,0.7,\) and \(0.3 .\) Experiment with other values of \(h\) What do you see happening as \(h \rightarrow 0 ?\) Explain this behavior.

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