Let's analyze the volumes of the sphere, cylinder, and cone based on the given dimensions:
-
Sphere: The volume \( V_s \) of a sphere with radius \( r \) is given by:
\[
V_s = \frac{4}{3} \pi r^3
\]
-
Cylinder: The volume \( V_c \) of a cylinder with radius \( r \) and height \( h \) is given by:
\[
V_c = \pi r^2 h
\]
Since the height \( h \) is twice the radius, we have \( h = 2r \). Thus, the volume of the cylinder becomes:
\[
V_c = \pi r^2 (2r) = 2\pi r^3
\]
-
Cone: The volume \( V_{cone} \) of a cone with radius \( r \) and height \( h \) is given by:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Again, using \( h = 2r \), we have:
\[
V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Now we can compare the volumes of the three shapes:
-
Cylinder ( \( V_c \) ):
\[
V_c = 2\pi r^3
\]
-
Cone ( \( V_{cone} \) ):
\[
V_{cone} = \frac{2}{3} \pi r^3
\]
-
Sphere ( \( V_s \) ):
\[
V_s = \frac{4}{3} \pi r^3
\]
Now let's express these in terms of a common volume unit, \( \pi r^3 \):
-
Volume of the sphere:
\[
V_s = \frac{4}{3} \pi r^3
\]
-
Volume of the cylinder:
\[
V_c = 2\pi r^3
\]
-
Volume of the cone:
\[
V_{cone} = \frac{2}{3} \pi r^3
\]
Now let's see if we can relate these volumes:
Comparing the volumes, we can express the cylinder's volume in terms of the cone's volume:
\[
V_c = 2 \times \text{(Volume of the cone)} = 2 \times \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3
\]
Thus, the cylinder has the same volume as the sphere \( V_s \):
So, we have:
\[
V_s = V_c
\]
Then, if we sum the cone's volume twice, we can express it as the cylinder:
\[
V_c = 2 \times V_{cone}
\]
Finally, using these relationships, we find:
\[
\text{Sphere} = \text{Cylinder} + \text{Cone}
\]
Rearranging gives:
\[
\text{Sphere} - \text{Cone} = \text{Cylinder}
\]
Therefore, the correct response is:
sphere - cone = cylinder.
