To find the volume of the sphere, we first need to determine the radius of the cone, which is also the radius of the sphere, and the height of the cone, which is equal to the diameter of the sphere.
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
We know that the volume of the cone is \( 253 \pi , \text{cm}^3 \). So, we have:
\[ \frac{1}{3} \pi r^2 h = 253 \pi \]
We can get rid of \( \pi \) (since it's not zero) from both sides:
\[ \frac{1}{3} r^2 h = 253 \]
Multiplying both sides by 3 gives:
\[ r^2 h = 759 \]
Now, we know that the height of the cone \( h \) is equal to the diameter of the sphere, which is \( 2r \). Therefore, we can substitute \( h \) with \( 2r \):
\[ r^2 (2r) = 759 \]
This simplifies to:
\[ 2r^3 = 759 \]
Dividing both sides by 2 gives:
\[ r^3 = 379.5 \]
Now to find the volume of the sphere, we use the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting \( r^3 \) into the formula:
\[ V = \frac{4}{3} \pi (379.5) \]
Calculating \( \frac{4}{3} \times 379.5 \):
\[ V = \frac{4 \times 379.5}{3} = \frac{1518}{3} = 506 , \pi , \text{cm}^3 \]
Among the choices given, \( 503 \pi , \text{cm}^3 \) is very close but not exact. It seems there might be a rounding or small error in the options provided, as our derived volume is \( 506 , \pi , \text{cm}^3 \).
Thus, based on the closest option given, the final answer is:
503 π cm³.