Question
Amanda wants to build a pool in her backyard. Describe a reasonable shape, dimensions and volume for the pool could build.
Question 3 options:
Hemisphere with a radius of 55 yards
Cone with a radius of 5 inches and a height of 8 feet
Rectangular Prism with a length of 8 feet, a width of 6 feet and a depth of 6 feet
Cylinder with a radius of 6 feet and a height of 1 mile
Question 3 options:
Hemisphere with a radius of 55 yards
Cone with a radius of 5 inches and a height of 8 feet
Rectangular Prism with a length of 8 feet, a width of 6 feet and a depth of 6 feet
Cylinder with a radius of 6 feet and a height of 1 mile
Answers
Answered by
GPT-4o mini
To determine a reasonable shape, dimensions, and volume for Amanda's pool, we can evaluate each option based on typical pool standards and practicality.
1. **Hemisphere with a radius of 55 yards**:
- **Volume**: The volume \( V \) of a hemisphere is given by the formula:
\[
V = \frac{2}{3} \pi r^3
\]
For a radius of 55 yards:
\[
V = \frac{2}{3} \pi (55)^3 \approx 7036.59 \text{ cubic yards}
\]
- This is a large volume, suitable for a backyard pool, but the dimensions and the spherical shape might not be practical for most residential properties.
2. **Cone with a radius of 5 inches and a height of 8 feet**:
- **Volume**: The volume of a cone is calculated with the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Converting the radius to feet (5 inches = \( \frac{5}{12} \) feet), we get:
\[
V = \frac{1}{3} \pi \left(\frac{5}{12}\right)^2 (8) \approx 5.27 \text{ cubic feet}
\]
- This is an impractically small volume for a pool.
3. **Rectangular Prism with a length of 8 feet, a width of 6 feet, and a depth of 6 feet**:
- **Volume**: The volume is calculated as:
\[
V = \text{length} \times \text{width} \times \text{depth} = 8 \times 6 \times 6 = 288 \text{ cubic feet}
\]
- This is a reasonable size for a small pool suitable for a backyard, especially for recreational use.
4. **Cylinder with a radius of 6 feet and a height of 1 mile**:
- **Volume**: The volume of a cylinder is given by:
\[
V = \pi r^2 h
\]
With a height converted to feet (1 mile = 5280 feet):
\[
V = \pi (6)^2 (5280) \approx 60137.93 \text{ cubic feet}
\]
- This is an absurdly large volume for a pool, making it impractical.
**Conclusion**:
The most reasonable option for Amanda's backyard pool is the **Rectangular Prism with a length of 8 feet, a width of 6 feet, and a depth of 6 feet**. It provides a practical volume and size for residential use.
1. **Hemisphere with a radius of 55 yards**:
- **Volume**: The volume \( V \) of a hemisphere is given by the formula:
\[
V = \frac{2}{3} \pi r^3
\]
For a radius of 55 yards:
\[
V = \frac{2}{3} \pi (55)^3 \approx 7036.59 \text{ cubic yards}
\]
- This is a large volume, suitable for a backyard pool, but the dimensions and the spherical shape might not be practical for most residential properties.
2. **Cone with a radius of 5 inches and a height of 8 feet**:
- **Volume**: The volume of a cone is calculated with the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Converting the radius to feet (5 inches = \( \frac{5}{12} \) feet), we get:
\[
V = \frac{1}{3} \pi \left(\frac{5}{12}\right)^2 (8) \approx 5.27 \text{ cubic feet}
\]
- This is an impractically small volume for a pool.
3. **Rectangular Prism with a length of 8 feet, a width of 6 feet, and a depth of 6 feet**:
- **Volume**: The volume is calculated as:
\[
V = \text{length} \times \text{width} \times \text{depth} = 8 \times 6 \times 6 = 288 \text{ cubic feet}
\]
- This is a reasonable size for a small pool suitable for a backyard, especially for recreational use.
4. **Cylinder with a radius of 6 feet and a height of 1 mile**:
- **Volume**: The volume of a cylinder is given by:
\[
V = \pi r^2 h
\]
With a height converted to feet (1 mile = 5280 feet):
\[
V = \pi (6)^2 (5280) \approx 60137.93 \text{ cubic feet}
\]
- This is an absurdly large volume for a pool, making it impractical.
**Conclusion**:
The most reasonable option for Amanda's backyard pool is the **Rectangular Prism with a length of 8 feet, a width of 6 feet, and a depth of 6 feet**. It provides a practical volume and size for residential use.