Question
Order these cones in order from greatest volume to the least volume.
A cone with a height of 8 cm and a radius of 3 cm
A cone with a height of 9 cm and a diameter of 12 cm
A cone with a side height of 13 m and a radius of 5 m
A cone with a height of 18 cm and a radius of 8 cm
A cone with a height of 21 cm and a radius of 3 cm
A cone with a height of 8 cm and a radius of 3 cm
A cone with a height of 9 cm and a diameter of 12 cm
A cone with a side height of 13 m and a radius of 5 m
A cone with a height of 18 cm and a radius of 8 cm
A cone with a height of 21 cm and a radius of 3 cm
Answers
GPT-4o mini
To order the cones by volume, we can use the formula for the volume of a cone:
\[
V = \frac{1}{3} \pi r^2 h
\]
Where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
We'll calculate the volume for each cone one by one.
1. **Cone 1**: Height = 8 cm, Radius = 3 cm
\[
V_1 = \frac{1}{3} \pi (3^2) (8) = \frac{1}{3} \pi (9)(8) = \frac{72}{3} \pi = 24\pi \, \text{cm}^3
\]
2. **Cone 2**: Height = 9 cm, Diameter = 12 cm (Radius = 6 cm)
\[
V_2 = \frac{1}{3} \pi (6^2) (9) = \frac{1}{3} \pi (36)(9) = \frac{324}{3} \pi = 108\pi \, \text{cm}^3
\]
3. **Cone 3**: Side height = 13 m and Radius = 5 m. We first need to find the height (h) using the Pythagorean theorem. The side height is the slant height \( l \), where:
\[
l^2 = r^2 + h^2 \implies 13^2 = 5^2 + h^2 \implies 169 = 25 + h^2 \implies h^2 = 144 \implies h = 12 \, \text{m}
\]
\[
V_3 = \frac{1}{3} \pi (5^2) (12) = \frac{1}{3} \pi (25)(12) = \frac{300}{3} \pi = 100\pi \, \text{m}^3
\]
4. **Cone 4**: Height = 18 cm, Radius = 8 cm
\[
V_4 = \frac{1}{3} \pi (8^2) (18) = \frac{1}{3} \pi (64)(18) = \frac{1152}{3} \pi = 384\pi \, \text{cm}^3
\]
5. **Cone 5**: Height = 21 cm, Radius = 3 cm
\[
V_5 = \frac{1}{3} \pi (3^2) (21) = \frac{1}{3} \pi (9)(21) = \frac{189}{3} \pi = 63\pi \, \text{cm}^3
\]
Now we can order the volumes:
- **Cone 4**: \( 384\pi \, \text{cm}^3 \)
- **Cone 2**: \( 108\pi \, \text{cm}^3 \)
- **Cone 3**: \( 100\pi \, \text{m}^3 \) (Note: Convert to cm³: \( 100\pi \, \text{m}^3 = 100000000\pi \, \text{cm}^3 \))
- **Cone 1**: \( 24\pi \, \text{cm}^3 \)
- **Cone 5**: \( 63\pi \, \text{cm}^3 \)
Ordering them from greatest volume to least:
1. **Cone 4**: \( 384\pi \, \text{cm}^3 \)
2. **Cone 2**: \( 108\pi \, \text{cm}^3 \)
3. **Cone 3**: \( 100000000\pi \, \text{cm}^3 \) (largest volume in m³, converted to cm³)
4. **Cone 5**: \( 63\pi \, \text{cm}^3 \)
5. **Cone 1**: \( 24\pi \, \text{cm}^3 \)
Final ordering (with Cone 4 as the largest):
1. Cone 4
2. Cone 2
3. Cone 3
4. Cone 5
5. Cone 1
\[
V = \frac{1}{3} \pi r^2 h
\]
Where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
We'll calculate the volume for each cone one by one.
1. **Cone 1**: Height = 8 cm, Radius = 3 cm
\[
V_1 = \frac{1}{3} \pi (3^2) (8) = \frac{1}{3} \pi (9)(8) = \frac{72}{3} \pi = 24\pi \, \text{cm}^3
\]
2. **Cone 2**: Height = 9 cm, Diameter = 12 cm (Radius = 6 cm)
\[
V_2 = \frac{1}{3} \pi (6^2) (9) = \frac{1}{3} \pi (36)(9) = \frac{324}{3} \pi = 108\pi \, \text{cm}^3
\]
3. **Cone 3**: Side height = 13 m and Radius = 5 m. We first need to find the height (h) using the Pythagorean theorem. The side height is the slant height \( l \), where:
\[
l^2 = r^2 + h^2 \implies 13^2 = 5^2 + h^2 \implies 169 = 25 + h^2 \implies h^2 = 144 \implies h = 12 \, \text{m}
\]
\[
V_3 = \frac{1}{3} \pi (5^2) (12) = \frac{1}{3} \pi (25)(12) = \frac{300}{3} \pi = 100\pi \, \text{m}^3
\]
4. **Cone 4**: Height = 18 cm, Radius = 8 cm
\[
V_4 = \frac{1}{3} \pi (8^2) (18) = \frac{1}{3} \pi (64)(18) = \frac{1152}{3} \pi = 384\pi \, \text{cm}^3
\]
5. **Cone 5**: Height = 21 cm, Radius = 3 cm
\[
V_5 = \frac{1}{3} \pi (3^2) (21) = \frac{1}{3} \pi (9)(21) = \frac{189}{3} \pi = 63\pi \, \text{cm}^3
\]
Now we can order the volumes:
- **Cone 4**: \( 384\pi \, \text{cm}^3 \)
- **Cone 2**: \( 108\pi \, \text{cm}^3 \)
- **Cone 3**: \( 100\pi \, \text{m}^3 \) (Note: Convert to cm³: \( 100\pi \, \text{m}^3 = 100000000\pi \, \text{cm}^3 \))
- **Cone 1**: \( 24\pi \, \text{cm}^3 \)
- **Cone 5**: \( 63\pi \, \text{cm}^3 \)
Ordering them from greatest volume to least:
1. **Cone 4**: \( 384\pi \, \text{cm}^3 \)
2. **Cone 2**: \( 108\pi \, \text{cm}^3 \)
3. **Cone 3**: \( 100000000\pi \, \text{cm}^3 \) (largest volume in m³, converted to cm³)
4. **Cone 5**: \( 63\pi \, \text{cm}^3 \)
5. **Cone 1**: \( 24\pi \, \text{cm}^3 \)
Final ordering (with Cone 4 as the largest):
1. Cone 4
2. Cone 2
3. Cone 3
4. Cone 5
5. Cone 1