The base of the pyramid is a regular hexagon. Find the volume of the pyramid. Round your answer to the nearest tenth. om 4 cm cm³
The base of the pyramid is a regular hexagon. Find the volume of the pyramid. Round your answer to the nearest tenth. om 4 cm cm³
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Volume Calculation of a Pyramid with a Regular Hexagon Base
**Problem Statement:**
The base of the pyramid is a regular hexagon. Find the volume of the pyramid. Round your answer to the nearest tenth.
**Given:**
- The base side length of the hexagon is 4 cm.
- The height (altitude) of the pyramid is to be calculated and inserted.
**Diagram Explanation:**
The accompanying diagram illustrates a pyramid with:
- A regular hexagon at its base.
- Each side of the hexagon measuring 4 cm.
- The height (H) from the apex to the center of the hexagonal base is indicated but not yet provided numerically.
The given values in the diagram are directly related to solving the volume of the pyramid, which will require finding the area of the hexagonal base and its height.
**Formula:**
1. **Calculate the area of the base (A):**
The area of a regular hexagon with side length \(a\) is given by:
\[
A = \frac{3\sqrt{3}a^2}{2}
\]
For \(a = 4 \, \text{cm}\):
\[
A = \frac{3\sqrt{3}(4)^2}{2} = \frac{3\sqrt{3}(16)}{2} = 24\sqrt{3} \, \text{cm}^2
\]
2. **Calculate the volume (V) of the pyramid:**
The volume \(V\) of a pyramid is given by:
\[
V = \frac{1}{3} \times A \times H
\]
Where \(A\) is the area of the base and \(H\) is the height.
Since \(H\) needs to be rounded to the nearest tenth in the final step, insert the given values, apply the calculations, and present the final result.
**Calculation and Rounding:**
1. Once \(H\) is determined/measured:
\[
H \, \text{cm}
\]
2. Substitute the values back into the volume formula:
\[
V = \frac{1}{3} \times 24\sqrt{3} \times H
\]
3. Perform the numerical operations and round the resulting volume to the nearest tenth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1658c097-87f5-41e9-88ac-7afef3364230%2F25e62dab-5ac5-46c7-ba32-3ea13eb63dbe%2Fvqdds5v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Volume Calculation of a Pyramid with a Regular Hexagon Base
**Problem Statement:**
The base of the pyramid is a regular hexagon. Find the volume of the pyramid. Round your answer to the nearest tenth.
**Given:**
- The base side length of the hexagon is 4 cm.
- The height (altitude) of the pyramid is to be calculated and inserted.
**Diagram Explanation:**
The accompanying diagram illustrates a pyramid with:
- A regular hexagon at its base.
- Each side of the hexagon measuring 4 cm.
- The height (H) from the apex to the center of the hexagonal base is indicated but not yet provided numerically.
The given values in the diagram are directly related to solving the volume of the pyramid, which will require finding the area of the hexagonal base and its height.
**Formula:**
1. **Calculate the area of the base (A):**
The area of a regular hexagon with side length \(a\) is given by:
\[
A = \frac{3\sqrt{3}a^2}{2}
\]
For \(a = 4 \, \text{cm}\):
\[
A = \frac{3\sqrt{3}(4)^2}{2} = \frac{3\sqrt{3}(16)}{2} = 24\sqrt{3} \, \text{cm}^2
\]
2. **Calculate the volume (V) of the pyramid:**
The volume \(V\) of a pyramid is given by:
\[
V = \frac{1}{3} \times A \times H
\]
Where \(A\) is the area of the base and \(H\) is the height.
Since \(H\) needs to be rounded to the nearest tenth in the final step, insert the given values, apply the calculations, and present the final result.
**Calculation and Rounding:**
1. Once \(H\) is determined/measured:
\[
H \, \text{cm}
\]
2. Substitute the values back into the volume formula:
\[
V = \frac{1}{3} \times 24\sqrt{3} \times H
\]
3. Perform the numerical operations and round the resulting volume to the nearest tenth.
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