[50 points] if the volume of a cylinder is 502400 cm what is the volume of a sphere if it has the same radius and height of the cylinder

The volume of the sphere would be 334933.33 cm³.

When comparing the formulas for the volume of a sphere and the volume of a cylinder, we think about the height of the sphere.  The radius of the sphere goes from the center to each outer edge; therefore the height of the sphere, running from top and bottom of the sphere, is 2 times the radius or 2r.

Formula for volume of a cylinder:
V = πr²h

Substituting 2r for h:

V = πr²(2r) = 2πr³

The formula for the volume of a sphere:

V = (4/3)πr³

Since 4/3 < 2, the sphere has a smaller volume.  The ratio of the volume of the sphere to the volume of the cylinder is 4/3:2, which can be written as:


This means that the volume of the sphere is 2/3 the volume of the cylinder.  We take the volume of the cylinder, 502400, and multiply it by 2/3:

334,933.33