The greenhouse pictured below can be modeled as a rectangular prism with a half-cylinder on top. The rectangular prism is 20 feet wide, 12 feet high, and 45 feet long. The half-cylinder has a diameter of 20 feet. To the nearest cubic foot, what is the volume of the greenhouse? 20 ft 45 ft 12 ft
The greenhouse pictured below can be modeled as a rectangular prism with a half-cylinder on top. The rectangular prism is 20 feet wide, 12 feet high, and 45 feet long. The half-cylinder has a diameter of 20 feet. To the nearest cubic foot, what is the volume of the greenhouse? 20 ft 45 ft 12 ft
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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![### Greenhouse Volume Calculation
**Problem:**
The greenhouse pictured below can be modeled as a rectangular prism with a half-cylinder on top. The rectangular prism is 20 feet wide, 12 feet high, and 45 feet long. The half-cylinder has a diameter of 20 feet. To the nearest cubic foot, what is the volume of the greenhouse?
**Diagram Description:**
The provided image depicts a greenhouse structure comprising a combination of two geometric shapes:
1. **Rectangular Prism:** It forms the base of the greenhouse. The dimensions of the prism are:
- **Width:** 20 feet
- **Height:** 12 feet
- **Length:** 45 feet
2. **Half-Cylinder:** This shape forms the curved roof of the greenhouse. The dimensions given are:
- **Diameter:** 20 feet (which implies a radius of 10 feet as the radius is half of the diameter)
- **Length:** 45 feet
**Steps to Calculate the Volume:**
1. **Volume of the Rectangular Prism:**
- The formula for the volume of a rectangular prism is:
\[
\text{Volume} = \text{Width} \times \text{Height} \times \text{Length}
\]
- Substituting the given dimensions:
\[
\text{Volume of Prism} = 20 \, \text{ft} \times 12 \, \text{ft} \times 45 \, \text{ft} = 10,800 \, \text{cubic feet}
\]
2. **Volume of the Half-Cylinder:**
- The formula for the volume of a cylinder is:
\[
\text{Volume} = \pi \times \text{radius}^2 \times \text{height}
\]
- Since the radius is 10 feet, and the height (or length) is 45 feet:
\[
\text{Volume of Cylinder} = \pi \times 10^2 \times 45 = 4,500\pi \, \text{cubic feet}
\]
- Since we have a half-cylinder, the volume is half of the calculated cylinder volume:
\[
\text{Volume of Half-Cylinder} = \frac{4,500\pi}{2} = 2,250](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa88d205-dd76-4939-b9bf-fa2466cfe1d2%2F6b37d256-ef2b-448c-836f-df973e10bbe8%2F3ptvp8m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Greenhouse Volume Calculation
**Problem:**
The greenhouse pictured below can be modeled as a rectangular prism with a half-cylinder on top. The rectangular prism is 20 feet wide, 12 feet high, and 45 feet long. The half-cylinder has a diameter of 20 feet. To the nearest cubic foot, what is the volume of the greenhouse?
**Diagram Description:**
The provided image depicts a greenhouse structure comprising a combination of two geometric shapes:
1. **Rectangular Prism:** It forms the base of the greenhouse. The dimensions of the prism are:
- **Width:** 20 feet
- **Height:** 12 feet
- **Length:** 45 feet
2. **Half-Cylinder:** This shape forms the curved roof of the greenhouse. The dimensions given are:
- **Diameter:** 20 feet (which implies a radius of 10 feet as the radius is half of the diameter)
- **Length:** 45 feet
**Steps to Calculate the Volume:**
1. **Volume of the Rectangular Prism:**
- The formula for the volume of a rectangular prism is:
\[
\text{Volume} = \text{Width} \times \text{Height} \times \text{Length}
\]
- Substituting the given dimensions:
\[
\text{Volume of Prism} = 20 \, \text{ft} \times 12 \, \text{ft} \times 45 \, \text{ft} = 10,800 \, \text{cubic feet}
\]
2. **Volume of the Half-Cylinder:**
- The formula for the volume of a cylinder is:
\[
\text{Volume} = \pi \times \text{radius}^2 \times \text{height}
\]
- Since the radius is 10 feet, and the height (or length) is 45 feet:
\[
\text{Volume of Cylinder} = \pi \times 10^2 \times 45 = 4,500\pi \, \text{cubic feet}
\]
- Since we have a half-cylinder, the volume is half of the calculated cylinder volume:
\[
\text{Volume of Half-Cylinder} = \frac{4,500\pi}{2} = 2,250
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