One hundred pyramid-shaped chocolate candies with a square base with 12 mm sides and height of 15 mm are melted in a cylindrical pot. If the pot has a radius of 75 mm, what is the height of the melted candies in the pot?
One hundred pyramid-shaped chocolate candies with a square base with 12 mm sides and height of 15 mm are melted in a cylindrical pot. If the pot has a radius of 75 mm, what is the height of the melted candies in the pot?
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![### Volume Calculation of Melted Pyramid-Shaped Candies
**Problem Statement:**
One hundred pyramid-shaped chocolate candies with a square base with 12 mm sides and height of 15 mm are melted in a cylindrical pot. If the pot has a radius of 75 mm, what is the height of the melted candies in the pot?
**Solution:**
1. **Calculate the volume of one pyramid-shaped candy:**
The formula for the volume of a pyramid with a square base is:
\( V = \frac{1}{3} \times \text{base area} \times \text{height} \)
- Base area (\( A \)) = side\(^2\)
\[ A = 12 \, \text{mm} \times 12 \, \text{mm} = 144 \, \text{mm}^2 \]
- Height (\( h \)) = 15 mm
- Volume (\( V \)) of one pyramid-shaped candy:
\[ V = \frac{1}{3} \times 144 \, \text{mm}^2 \times 15 \, \text{mm} \]
\[ V = \frac{1}{3} \times 2160 \, \text{mm}^3 \]
\[ V = 720 \, \text{mm}^3 \]
2. **Calculate the total volume of 100 pyramid-shaped candies:**
- Total volume:
\[ V_{\text{total}} = 100 \times 720 \, \text{mm}^3 \]
\[ V_{\text{total}} = 72,000 \, \text{mm}^3 \]
3. **Calculate the height of the melted candies in the cylindrical pot:**
The formula for the volume of a cylinder is:
\[ V = \pi r^2 h \]
- Given:
- Radius (\( r \)) = 75 mm
- Total volume (\( V_{\text{total}} \)) = 72,000 mm\(^3\)
- Height (\( h \)) is unknown. Rearrange the volume formula to solve for \( h \):
\[ h = \frac{V}{\pi r^2} \]
\[ h = \frac{72,000](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a429899-9f5d-4623-89a9-172424948446%2F82298025-c28f-4084-89bb-3d537f77b438%2F25ez2gf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Volume Calculation of Melted Pyramid-Shaped Candies
**Problem Statement:**
One hundred pyramid-shaped chocolate candies with a square base with 12 mm sides and height of 15 mm are melted in a cylindrical pot. If the pot has a radius of 75 mm, what is the height of the melted candies in the pot?
**Solution:**
1. **Calculate the volume of one pyramid-shaped candy:**
The formula for the volume of a pyramid with a square base is:
\( V = \frac{1}{3} \times \text{base area} \times \text{height} \)
- Base area (\( A \)) = side\(^2\)
\[ A = 12 \, \text{mm} \times 12 \, \text{mm} = 144 \, \text{mm}^2 \]
- Height (\( h \)) = 15 mm
- Volume (\( V \)) of one pyramid-shaped candy:
\[ V = \frac{1}{3} \times 144 \, \text{mm}^2 \times 15 \, \text{mm} \]
\[ V = \frac{1}{3} \times 2160 \, \text{mm}^3 \]
\[ V = 720 \, \text{mm}^3 \]
2. **Calculate the total volume of 100 pyramid-shaped candies:**
- Total volume:
\[ V_{\text{total}} = 100 \times 720 \, \text{mm}^3 \]
\[ V_{\text{total}} = 72,000 \, \text{mm}^3 \]
3. **Calculate the height of the melted candies in the cylindrical pot:**
The formula for the volume of a cylinder is:
\[ V = \pi r^2 h \]
- Given:
- Radius (\( r \)) = 75 mm
- Total volume (\( V_{\text{total}} \)) = 72,000 mm\(^3\)
- Height (\( h \)) is unknown. Rearrange the volume formula to solve for \( h \):
\[ h = \frac{V}{\pi r^2} \]
\[ h = \frac{72,000
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