Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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**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

![Diagram of a cylindrical garden compost bin](image-of-compos-bin.jpg)

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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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 ![Diagram of a cylindrical garden compost bin](image-of-compos-bin.jpg) 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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