To analyze the relationships between the volumes of the sphere, cylinder, and cone with the given conditions, let's calculate their volumes first:
-
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 \( h = 2r \): \[ V_c = \pi r^2 (2r) = 2\pi r^3 \]
-
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, substituting \( h = 2r \): \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we have:
- 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: \( V_{cone} = \frac{2}{3} \pi r^3 \)
Next, we can analyze the relationships:
-
Is the volume of the sphere equal to the volume of the cylinder plus the cone? \[ V_c + V_{cone} = 2\pi r^3 + \frac{2}{3}\pi r^3 \] \[ = \frac{6}{3}\pi r^3 + \frac{2}{3}\pi r^3 = \frac{8}{3}\pi r^3 \] This is not equal to \( V_s \).
-
Is the volume of the cylinder equal to 2 cones plus the sphere? \[ 2 V_{cone} + V_s = 2 \left(\frac{2}{3}\pi r^3\right) + \frac{4}{3}\pi r^3 = \frac{4}{3}\pi r^3 + \frac{4}{3}\pi r^3 = \frac{8}{3}\pi r^3 \] Again, this is not equal to \( V_c \).
-
Is the volume of the sphere minus the cone equal to the cylinder? \[ V_s - V_{cone} = \frac{4}{3}\pi r^3 - \frac{2}{3}\pi r^3 = \frac{2}{3}\pi r^3 \] This is not equal to \( V_c \).
-
Is three cones equal to one cylinder? \[ 3 V_{cone} = 3 \left(\frac{2}{3}\pi r^3\right) = 2\pi r^3 \] This is exactly equal to \( V_c \).
Thus, the true statement is: \[ 3 \text{ cones} = 1 \text{ cylinder} \]