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

1. sphere = cylinder + cone
2. 3 cones = 1 cylinder
3. 1 cylinder = 2 cones + 1 sphere
4. sphere - cone = cylinder

1 answer

To analyze the relationships between the volumes of a sphere, cylinder, and cone, we can start with the volume formulas for each shape:

  1. Volume of the sphere (V_s): \[ V_s = \frac{4}{3} \pi r^3 \] where \( r \) is the radius.

  2. Volume of the cylinder (V_c): \[ V_c = \pi r^2 h \] where \( h \) is the height and \( r \) is the radius.

  3. Volume of the cone (V_{cone}): \[ V_{cone} = \frac{1}{3} \pi r^2 h \]

If we assume that the cone and cylinder have the same height \( h \) and share the same radius \( r \):

Now let's evaluate the statements:

  1. sphere = cylinder + cone \[ \frac{4}{3} \pi r^3 \neq \pi r^2 h + \frac{1}{3} \pi r^2 h \] (Not true, as the left-hand side scales with \( r^3 \) while the right scales with \( r^2 \) and \( h \)).

  2. 3 cones = 1 cylinder \[ 3 \left( \frac{1}{3} \pi r^2 h \right) = \pi r^2 h \] (True: The volumes are equal).

  3. 1 cylinder = 2 cones + 1 sphere \[ \pi r^2 h \neq 2 \left( \frac{1}{3} \pi r^2 h \right) + \frac{4}{3} \pi r^3 \] (Not true, as the dimensions are inconsistent).

  4. sphere - cone = cylinder \[ \frac{4}{3} \pi r^3 - \left( \frac{1}{3} \pi r^2 h \right) \neq \pi r^2 h \] (Not true, again inconsistencies in terms).

Conclusion:

The only true statement from the options presented is:

2. 3 cones = 1 cylinder.