SOLUTION: The largest sphere that will fit is placed inside a cone with base radius 𝑟 and height ℎ. What proportion of the cone is filled by the sphere?

First we look at the mid cross section.
Let the radius of the circle (which is also the radius of the sphere),
be "a".



Then the ratio of the volume of the sphere to the volume of the cone,
which is the answer you're looking for, will be



The hard part is finding the radius of the sphere "a", in terms
of "r" and "h".

By similar right triangles OPQ and TSQ,



SQ = h
SO = a
OQ = SQ-SO = h-a
OP = a
TS = r


Substituting,

 

Cross-multiplying:

  

 






 
Now we substitute that horrible thing in: 





Edwin

Now the above problem is correct!

Edwin