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Chapter 3: Problem 43

Estimating volume Estimate the volume of material in a cylindrical shell with length 30 in. radius 6 in. and shell thickness 0.5 in.

Short Answer

Expert verified
The volume of material in the shell is approximately 541.12 cubic inches.

Step by step solution

01

Understand the Shape

We are dealing with a cylindrical shell, which means it is a hollow cylinder. The problem provides the outer radius (6 in) and shell thickness (0.5 in). The cylinder's length is 30 in.
02

Identify Inner and Outer Cylinders

The shell thickness is given as 0.5 in. Therefore, the inner radius of the shell can be calculated as the outer radius minus the thickness of the shell, which is 6 in - 0.5 in = 5.5 in.
03

Calculate Outer Volume

The volume of a cylinder is calculated using the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. For the outer cylinder, use the outer radius (6 in) and length (30 in): \[ V_{outer} = \pi (6^2) (30) = 1080\pi \text{ cubic inches} \].
04

Calculate Inner Volume

Apply the same formula \( V = \pi r^2 h \) using the inner radius (5.5 in) and length (30 in): \[ V_{inner} = \pi (5.5^2) (30) = 907.5\pi \text{ cubic inches} \].
05

Calculate Volume of Material in the Shell

The volume of material in the cylindrical shell is the difference between the outer and inner volumes: \[ V_{material} = V_{outer} - V_{inner} = 1080\pi - 907.5\pi = 172.5\pi \text{ cubic inches} \].
06

Approximate Value

To get a numerical approximation, use \( \pi \approx 3.14159 \) and calculate \[ 172.5 \times 3.14159 \approx 541.12 \text{ cubic inches} \].

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Key Concepts

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

Cylindrical Shell
A cylindrical shell is essentially a hollow cylindrical structure. Imagine a long tube, similar to a pipe, but open at both ends. The cylindrical shell geometry consists of two circular components: an inner cylinder and an outer cylinder. These cylinders share the same central axis, but they have different radii. The thickness of the shell is the difference between the outer and inner radii. This structure is often used in contexts where material efficiency and structural strength are important.
  • Inner and Outer Cylinders: Different radius values.
  • Shell Thickness: Determines the volume contained in the material.
  • Commonly Used in Engineering: Pipes, storage tanks, and silos.
Volume Calculation
When calculating the volume of a cylindrical shell, you are essentially finding out how much space the material of the shell occupies. A cylindrical shell's volume is determined by calculating the difference in volume between the outer cylinder and the inner cylinder. This process ensures that only the volume of the material itself is measured.The general formula for the volume of a cylinder is given by:\[ V = \pi r^2 h \]where:
  • \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
By applying this formula to both the inner and outer cylinders, and subtracting their volumes, you obtain the volume of the material in the shell. This step-by-step approach ensures correct volume measurement between two layers.
Outer Volume
The outer volume of a cylindrical shell refers to the volume enclosed by the outermost surface of the cylinder. It is the volume of an imaginary solid cylinder that encapsulates the entire hollow structure if it were filled.To find the outer volume:
  • Use the formula \( V = \pi r^2 h \).
  • For our example, the outer radius is 6 inches and the height (or length) is 30 inches.
  • Thus, the outer volume calculates to \[ V_{outer} = \pi (6^2) (30) = 1080\pi \] cubic inches.
This calculation estimates the space confined in a full cylinder with the given radius and height. It's essential for determining the shell's total capacity if hollowed material wasn't considered.
Inner Volume
The inner volume represents the volume occupied by the inside space of the cylindrical shell. Imagine the hollow center of the cylinder; this is effectively what the inner volume outlines.To calculate the inner volume:
  • The same volume formula \( V = \pi r^2 h \) is applied again.
  • Here, the inner radius is found by subtracting the shell's thickness from the outer radius. Hence, the inner radius is 5.5 inches.
  • Given the height as 30 inches, the equation is \[ V_{inner} = \pi (5.5^2) (30) = 907.5\pi \] cubic inches.
This calculation shows the void space and is subtracted from the outer volume in order to isolate the volume material in the shell itself.

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