A garden compost bin on sale at the hardware store is shaped like a cylinder with a 40 inch height and 18 inch diameter. 40 in 18 in To allow the compost to be mixed, the manufacturer recommends that it not be filled more than 2 full. What is the maximum recommended amount of compost that can be added to the bin, to the near A.O 6, 786 in³ B.O 27, 143 in³ c.O 2, 771 in D.O 10, 179 in3
A garden compost bin on sale at the hardware store is shaped like a cylinder with a 40 inch height and 18 inch diameter. 40 in 18 in To allow the compost to be mixed, the manufacturer recommends that it not be filled more than 2 full. What is the maximum recommended amount of compost that can be added to the bin, to the near A.O 6, 786 in³ B.O 27, 143 in³ c.O 2, 771 in D.O 10, 179 in3
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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Question
To ensure proper mixing of the compost, the manufacturer recommends filling the bin no more than 2/3 full. The goal is to determine the maximum recommended amount of compost that can be added to this bin, expressed in cubic inches to the nearest whole number.
### Step-by-Step Calculation:
1. **Calculate the Radius**: The radius of the cylindrical bin is half of the diameter.
\[
\text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{18 \text{ inches}}{2} = 9 \text{ inches}
\]
2. **Volume of the Cylinder**: The volume \(V\) of a cylinder is calculated using the formula:
\[
V = \pi r^2 h
\]
where \(r\) is the radius and \(h\) is the height.
\[
V = \pi (9 \text{ inches})^2 (40 \text{ inches}) = \pi \times 81 \text{ in}^2 \times 40 \text{ in} = 3240\pi \text{ in}^3
\]
3. **Approximate the Volume**: Using \(\pi \approx 3.1416\),
\[
V \approx 3240 \times 3.1416 \approx 10,179 \text{ cubic inches}
\]
4. **Maximum Recommended Amount of Compost**: Since the bin should only be filled to 2/3 of its full volume:
\[
\text{Recommended Volume} = \frac{2}{3} \times 10,179 \approx 6,786 \text{ cubic inches}
\]
### Answer
After performing the calculations, the maximum recommended amount of compost that can be added to the bin—to the nearest cubic inch—is:
\[
\boxed{6,786 \text{ in}^3}
\]
### Multiple-Choice Options:
- A. 6,786 in³
- B. 27,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb602927-d5c2-4b17-99bc-cf43c066f5a5%2Fe5e7bfa8-87d9-4849-819f-0671fa4c54d1%2Fawwb4a8_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Volume Capacity of a Garden Compost Bin**
A garden compost bin on sale at the hardware store is shaped like a cylinder. Here are its dimensions:
- **Height**: 40 inches
- **Diameter**: 18 inches
To ensure proper mixing of the compost, the manufacturer recommends filling the bin no more than 2/3 full. The goal is to determine the maximum recommended amount of compost that can be added to this bin, expressed in cubic inches to the nearest whole number.
### Step-by-Step Calculation:
1. **Calculate the Radius**: The radius of the cylindrical bin is half of the diameter.
\[
\text{Radius} (r) = \frac{\text{Diameter}}{2} = \frac{18 \text{ inches}}{2} = 9 \text{ inches}
\]
2. **Volume of the Cylinder**: The volume \(V\) of a cylinder is calculated using the formula:
\[
V = \pi r^2 h
\]
where \(r\) is the radius and \(h\) is the height.
\[
V = \pi (9 \text{ inches})^2 (40 \text{ inches}) = \pi \times 81 \text{ in}^2 \times 40 \text{ in} = 3240\pi \text{ in}^3
\]
3. **Approximate the Volume**: Using \(\pi \approx 3.1416\),
\[
V \approx 3240 \times 3.1416 \approx 10,179 \text{ cubic inches}
\]
4. **Maximum Recommended Amount of Compost**: Since the bin should only be filled to 2/3 of its full volume:
\[
\text{Recommended Volume} = \frac{2}{3} \times 10,179 \approx 6,786 \text{ cubic inches}
\]
### Answer
After performing the calculations, the maximum recommended amount of compost that can be added to the bin—to the nearest cubic inch—is:
\[
\boxed{6,786 \text{ in}^3}
\]
### Multiple-Choice Options:
- A. 6,786 in³
- B. 27,
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