You have a cylinder with a diameter of 15 inches and a height of 12 inches. what is the volume of the largest sphere that will fit inside it? if you can will you show your work somehow?

The work is easy.  Most of the effort is in the reasoning.
Here's how I reasoned:

-- If the diameter of the sphere is bigger than the diameter of
the cylinder, then you'd have to mash the sphere somehow 
in order to get it in there, and it wouldn't be a sphere anymore.

-- If the diameter of the sphere is bigger than the height of the
cylinder, then it would stick out of the ends, and it would not
"fit inside it".

-- So the diameter of the sphere can't be bigger than either
the diameter or the height of the cylinder.

-- The cylinder's diameter is 15" and its height is 12".
So the largest sphere that can completely fit inside it
is a sphere whose diameter is 12".

The volume of a sphere is  (4/3) (pi) (radius³)
and the radius is (1/2 of the diameter) = 6 inches.

So the volume is  (4/3) (pi) (6 inches)³ .

You can take it from here.
Use  3.14  for  (pi) .

I got about 904 inches³.
But the answer is not important.

Wow. that is a really cool question.