A grain silo can be modeled as a right cylinder topped with a hemisphere. Find the volume of the silo if it has a height of 48 m and a radius of 9 m. Round your answer to the nearest tenth if necessary. (Note: diagram is not drawn to scale.) 48 9
A grain silo can be modeled as a right cylinder topped with a hemisphere. Find the volume of the silo if it has a height of 48 m and a radius of 9 m. Round your answer to the nearest tenth if necessary. (Note: diagram is not drawn to scale.) 48 9
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Volume of a Grain Silo**
A grain silo can be modeled as a right cylinder topped with a hemisphere. Find the volume of the silo if it has a height of 48 m and a radius of 9 m. Round your answer to the nearest tenth if necessary. (Note: diagram is not drawn to scale.)
**Diagram Explanation:**
The diagram consists of a right cylinder with a hemispherical top. The cylinder has a height of 48 meters and a radius of 9 meters. The radius is consistent for both the cylindrical part and the hemispherical top.
**Solution:**
To find the total volume of the grain silo, we need to sum the volume of the cylindrical part and the volume of the hemispherical top.
1. **Volume of the Cylinder:**
The volume \(V_{cylinder}\) of a cylinder is calculated using the formula:
\[
V_{cylinder} = \pi r^2 h
\]
- \(r\) is the radius (9 m)
- \(h\) is the height (48 m)
\[
V_{cylinder} = \pi (9^2)(48) = \pi (81)(48) = 3888\pi \, \text{m}^3
\]
2. **Volume of the Hemisphere:**
The volume \(V_{hemisphere}\) of a hemisphere is calculated as half the volume of a sphere, using the formula:
\[
V_{hemisphere} = \frac{1}{2} \left( \frac{4}{3}\pi r^3 \right) = \frac{2}{3}\pi r^3
\]
- \(r\) is the radius (9 m)
\[
V_{hemisphere} = \frac{2}{3} \pi (9^3) = \frac{2}{3} \pi (729) = 486\pi \, \text{m}^3
\]
3. **Total Volume:**
The total volume \(V_{total}\) of the silo is the sum of the volumes of the cylinder and the hemisphere:
\[
V_{total} = V_{cylinder} + V_{hemisphere} = 3888\pi + 486\pi = 4374\pi \, \text{m}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98e632c3-d5e7-4ac9-8326-96ab02a3b3f6%2F316cad5e-cbf4-45f5-ab5f-6c74663766b1%2Fasqqi6c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Volume of a Grain Silo**
A grain silo can be modeled as a right cylinder topped with a hemisphere. Find the volume of the silo if it has a height of 48 m and a radius of 9 m. Round your answer to the nearest tenth if necessary. (Note: diagram is not drawn to scale.)
**Diagram Explanation:**
The diagram consists of a right cylinder with a hemispherical top. The cylinder has a height of 48 meters and a radius of 9 meters. The radius is consistent for both the cylindrical part and the hemispherical top.
**Solution:**
To find the total volume of the grain silo, we need to sum the volume of the cylindrical part and the volume of the hemispherical top.
1. **Volume of the Cylinder:**
The volume \(V_{cylinder}\) of a cylinder is calculated using the formula:
\[
V_{cylinder} = \pi r^2 h
\]
- \(r\) is the radius (9 m)
- \(h\) is the height (48 m)
\[
V_{cylinder} = \pi (9^2)(48) = \pi (81)(48) = 3888\pi \, \text{m}^3
\]
2. **Volume of the Hemisphere:**
The volume \(V_{hemisphere}\) of a hemisphere is calculated as half the volume of a sphere, using the formula:
\[
V_{hemisphere} = \frac{1}{2} \left( \frac{4}{3}\pi r^3 \right) = \frac{2}{3}\pi r^3
\]
- \(r\) is the radius (9 m)
\[
V_{hemisphere} = \frac{2}{3} \pi (9^3) = \frac{2}{3} \pi (729) = 486\pi \, \text{m}^3
\]
3. **Total Volume:**
The total volume \(V_{total}\) of the silo is the sum of the volumes of the cylinder and the hemisphere:
\[
V_{total} = V_{cylinder} + V_{hemisphere} = 3888\pi + 486\pi = 4374\pi \, \text{m}
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