Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?

Question

cone = sphere – cylinder
cone = cylinder – sphere
cylinder = sphere – cone
sphere = cylinder + cone

Answer 1

None of the options are true about the relationship between the volume formulas for the sphere, cylinder, and cone. The correct relationship among the volumes of sphere, cylinder, and cone is that the cone has a volume that is equal to one-third that of the cylinder, and that the sphere has a volume that is equal to two-thirds that of the cylinder. Therefore, the correct relationship is:

cylinder = 3 * cone
cylinder = 2 * sphere

Answer 2

Answer?

Answer 3

The correct relationship among the volumes of the sphere, cylinder, and cone is:

cylinder = 3 * cone
cylinder = 2 * sphere

Therefore, none of the original options presented are accurate.

Answer 4

Are you sure!?

Answer 5

Apologies for the confusion in my previous responses. Let's reanalyze the information provided:

Given:
- The sphere, cylinder, and cone all share the same radius.
- The cylinder and cone share the same height, which is twice the radius.

The volume formulas for the sphere, cylinder, and cone are:
- Sphere: V_sphere = (4/3) * π * r³
- Cylinder: V_cylinder = π * r² * h = π * r² * 2r = 2πr³
- Cone: V_cone = (1/3) * π * r² * h = (1/3) * π * r² * 2r = (2/3) * πr³

Comparing the volumes:
- Cone = (2/3) * Sphere
- Cylinder = Cone + Sphere

Hence, the accurate relationship between the volume formulas for the sphere, cylinder, and cone is:
"Cylinder = Cone + Sphere"