Put the ones in order from greatest volume (put at the top of the list) to least volume (put a the bottom of the list).

Question

Try to put them i order thinking about the forula instead of actually solving for the volume.
Responces:
A. The image depicts a geometric cone with labeled dimensions. The cone's slant height is indicated as 13 meters, its radius as 5 meters, and its height is not explicitly given but is a perpendicular line from the apex to the center of the circular base. The cone is outlined in blue, with the dimensions labeled in red.
B. The image shows a 3D rendering of a right circular cone, with labels indicating its dimensions. The cone is oriented vertically with its apex pointing upwards. The radius of the circular base is labeled as "4 cm" and the height of the cone, measured from the center of the base to the apex, is labeled as "15 cm". The cone is depicted with a faint orange outline and a dotted line representing the height inside. The background is plain white.
C. The image shows a geometric figure, specifically a right circular cone, with labeled dimensions. The height of the cone is labeled as 8 cm, and the radius of the circular base is labeled as 3 cm. This representation is typical of diagrams used in geometry to illustrate and solve problems involving the volume or surface area of cones.
D.The image depicts a green cone with measurements labeled. The height of the cone is labeled as 9 cm, and the diameter of the circular base is labeled as 12 cm. This implies that the radius of the base is 6 cm. The cone is oriented vertically, with the apex pointing upwards.
E. The image shows a purple cone with labeled dimensions. The radius (r) is labeled as 8 cm, the height (h) is labeled as 15 cm, and the slant height is labeled as 17 cm. The cone is positioned on its circular base, with its apex pointing upwards.
Responces:
A. The image depicts a geometric cone with labeled dimensions. The cone's slant height is indicated as 13 meters, its radius as 5 meters, and its height is not explicitly given but is a perpendicular line from the apex to the center of the circular base. The cone is outlined in blue, with the dimensions labeled in red.
B. The image shows a 3D rendering of a right circular cone, with labels indicating its dimensions. The cone is oriented vertically with its apex pointing upwards. The radius of the circular base is labeled as "4 cm" and the height of the cone, measured from the center of the base to the apex, is labeled as "15 cm". The cone is depicted with a faint orange outline and a dotted line representing the height inside. The background is plain white.
C. The image shows a geometric figure, specifically a right circular cone, with labeled dimensions. The height of the cone is labeled as 8 cm, and the radius of the circular base is labeled as 3 cm. This representation is typical of diagrams used in geometry to illustrate and solve problems involving the volume or surface area of cones.
D.The image depicts a green cone with measurements labeled. The height of the cone is labeled as 9 cm, and the diameter of the circular base is labeled as 12 cm. This implies that the radius of the base is 6 cm. The cone is oriented vertically, with the apex pointing upwards.
E. The image shows a purple cone with labeled dimensions. The radius (r) is labeled as 8 cm, the height (h) is labeled as 15 cm, and the slant height is labeled as 17 cm. The cone is positioned on its circular base, with its apex pointing upwards.

Answer 1

To order the cones from greatest volume to least volume, we need to consider the volume formula for a cone, which is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

where \( r \) is the radius and \( h \) is the height. We'll assess each option by calculating the values of \( r^2 \) and highlighting the corresponding heights:

1. E: Radius = 8 cm, Height = 15 cm
   \[ r^2 = 8^2 = 64, \quad h = 15 \] Volume factor = \( 64 \times 15 = 960 \)

2. B: Radius = 4 cm, Height = 15 cm
   \[ r^2 = 4^2 = 16, \quad h = 15 \] Volume factor = \( 16 \times 15 = 240 \)

3. D: Radius = 6 cm (diameter 12 cm), Height = 9 cm
   \[ r^2 = 6^2 = 36, \quad h = 9 \] Volume factor = \( 36 \times 9 = 324 \)

4. C: Radius = 3 cm, Height = 8 cm
   \[ r^2 = 3^2 = 9, \quad h = 8 \] Volume factor = \( 9 \times 8 = 72 \)

5. A: Radius = 5 meters, Height is unknown (but slant height is 13 meters)
   We can't determine volume precisely without the height. However, slant height must be √(r² + h²), which suggests it wouldn't exceed E due to constraints on r and typical dimensions.

Placing the cones in order of volume from greatest to least based on the calculations and evidence:

1. E: (Volume Factor = 960)
2. B: (Volume Factor = 240)
3. D: (Volume Factor = 324)
4. C: (Volume Factor = 72)
5. A: (Volume not definitively calculable, but likely less than C)

Final order:

1. E
2. B
3. D
4. C
5. A