To find the correct relationship between the volumes of the sphere, cylinder, and cone when they share the same radius (let's denote it as \( r \)) and the cylinder and cone share the same height (denote it as \( h = 2r \)), we need to calculate the volumes of each shape.
-
Volume of the Sphere (V_s): \[ V_s = \frac{4}{3} \pi r^3 \]
-
Volume of the Cylinder (V_c): \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
-
Volume of the Cone (V_{cone}): \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now let's examine the relationships:
- We already have: \[ V_c = 2\pi r^3 \] \[ V_{cone} = \frac{2}{3} \pi r^3 \]
Next, we can combine the volumes of the cone and sphere:
Adding the volume of the cone to the volume of the sphere: \[ V_s + V_{cone} = \frac{4}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 = 2\pi r^3 = V_c \]
Thus, the relationship established is: \[ V_s + V_{cone} = V_c \] or rearranging, \[ V_c = V_s + V_{cone} \]
This leads to the conclusion that:
- Cylinder = Sphere + Cone.
Therefore, the correct response is:
sphere = cylinder + cone