To find the volume of the sphere based on the provided information about the cone, we will begin by using the formula for the volume of a cone, which is:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.
Given that the volume of the cone is \( 253 \pi \), we can set up the equation:
\[ \frac{1}{3} \pi r^2 h = 253 \pi \]
Dividing both sides by \( \pi \):
\[ \frac{1}{3} r^2 h = 253 \]
Now, since the height \( h \) of the cone is equal to the diameter of the sphere and the radius of the sphere is equal to the radius of the cone, we have:
- Radius of sphere \( r_s = r \)
- Diameter of sphere \( d_s = 2r \) (which is equal to height \( h \) of the cone)
Now substituting \( h \) as \( 2r \) into the volume equation:
\[ \frac{1}{3} r^2 (2r) = 253 \]
This simplifies to:
\[ \frac{2}{3} r^3 = 253 \]
To find \( r^3 \):
\[ r^3 = 253 \cdot \frac{3}{2} = \frac{759}{2} \]
Now, we can find the volume of the sphere using the formula for the volume of a sphere:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
Substituting \( r^3 = \frac{759}{2} \):
\[ V_{sphere} = \frac{4}{3} \pi \cdot \frac{759}{2} \]
This can be simplified as:
\[ V_{sphere} = \frac{4 \cdot 759}{6} \pi = \frac{3036}{6} \pi = 506 \pi \]
Thus, the volume of the sphere is:
\[ \text{Volume of the sphere} = 506 \pi , \text{cm}^3 \]
Since this answer does not align with the options presented, it seems that there might be an inconsistency or mistake in the original volume of the cone or the statements provided. Would you like to reassess any of the provided values?