Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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### Diagram Description:
The diagram depicts an annular cylindrical shape. The measurements provided are:
- The outer radius (R) of the larger cylinder: **9 meters**.
- The height (h) of the cylinder: **5 meters**.
- The inner radius (r) of the removed cylinder: **3 meters**.
### Steps to Calculate the Volume:
1. **Calculate the Volume of the Outer Cylinder:**
\[
V_{\text{outer}} = \pi \cdot R^2 \cdot h
\]
Substituting the values:
\[
V_{\text{outer}} = \pi \cdot (9)^2 \cdot 5 = \pi \cdot 81 \cdot 5 = 405\pi \, \text{m}^3
\]
2. **Calculate the Volume of the Inner Cylinder:**
\[
V_{\text{inner}} = \pi \cdot r^2 \cdot h
\]
Substituting the values:
\[
V_{\text{inner}} = \pi \cdot (3)^2 \cdot 5 = \pi \cdot 9 \cdot 5 = 45\pi \, \text{m}^3
\]
3. **Determine the Volume of the Composite Solid (Annular Cylinder):**
This is achieved by subtracting the volume of the inner cylinder from the volume of the outer cylinder:
\[
V_{\text{composite}} = V_{\text{outer}} - V_{\text{inner}}
\]
Substituting our previous results:
\[
V_{\text{composite}} = 405\pi - 45\pi = 360\pi \, \text{m}^3
\]
Thus, the volume of the composite solid is:
\[
\boxed{360\pi \, \text{m}^3}
\]
or approximately:
\[
1130](https://content.bartleby.com/qna-images/question/fbe731da-e8f7-4d10-8184-315df05708b0/9f8da4eb-c0ae-4e2b-b083-9ac1ca2a1d82/52tm0qg_processed.jpeg)
Transcribed Image Text:**Calculating the Volume of a Composite Solid**
In this exercise, we will determine the volume of a composite solid structure. The structure in question comprises two cylindrical volumes: one larger cylinder with a smaller cylinder removed from its center, resulting in a ring or annular cylinder.
To follow this, observe the diagram below:
### Diagram Description:
The diagram depicts an annular cylindrical shape. The measurements provided are:
- The outer radius (R) of the larger cylinder: **9 meters**.
- The height (h) of the cylinder: **5 meters**.
- The inner radius (r) of the removed cylinder: **3 meters**.
### Steps to Calculate the Volume:
1. **Calculate the Volume of the Outer Cylinder:**
\[
V_{\text{outer}} = \pi \cdot R^2 \cdot h
\]
Substituting the values:
\[
V_{\text{outer}} = \pi \cdot (9)^2 \cdot 5 = \pi \cdot 81 \cdot 5 = 405\pi \, \text{m}^3
\]
2. **Calculate the Volume of the Inner Cylinder:**
\[
V_{\text{inner}} = \pi \cdot r^2 \cdot h
\]
Substituting the values:
\[
V_{\text{inner}} = \pi \cdot (3)^2 \cdot 5 = \pi \cdot 9 \cdot 5 = 45\pi \, \text{m}^3
\]
3. **Determine the Volume of the Composite Solid (Annular Cylinder):**
This is achieved by subtracting the volume of the inner cylinder from the volume of the outer cylinder:
\[
V_{\text{composite}} = V_{\text{outer}} - V_{\text{inner}}
\]
Substituting our previous results:
\[
V_{\text{composite}} = 405\pi - 45\pi = 360\pi \, \text{m}^3
\]
Thus, the volume of the composite solid is:
\[
\boxed{360\pi \, \text{m}^3}
\]
or approximately:
\[
1130
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