Question 1017561
<pre>
Here is a drawing of the mid-cross section of the 
largest sphere in the cone, which is the largest 
circle inside triangle ABC:

{{{drawing(400,2000/7,-7,7,-1,9,
locate(0.2,6,8-r),locate(-3.8,0,6),
locate(-6,0,A), locate(6,0,B),locate(-.2,8.5,C),
locate(-6,6,AC=10),
locate(-2.9,5,E), locate(0,0,D),locate(.2,3.2,O),
arc(0,3,6,-6,120,420),
locate(.1,1.7,r), locate(-1.1,4.5,r),
triangle(-6,0,6,0,0,8),  line(0,3,2.4,4.8), line(0,3,-2.4,4.8),
line(0,8,0,0) )}}}

Right triangles ADC and OEC are similar because they share
a common angle at A.

The hypotenuse of ADC is AC.  And AC = 10 by the Pythagorean 
theorem applied to triangle ADC. (6²+8²=10²)

{{{(OC)/(AC)}}}{{{""=""}}}{{{OE/AD}}}

So:

{{{(8-r)/10}}}{{{""=""}}}{{{r/6}}}

Cross-multiply and solve that and get r = 3

So the radius of the circle is 3, which is also the radius of
the sphere.

The volume of a sphere is given by the formula

{{{V}}}{{{""=""}}}{{{expr(4/3)pi*r^3}}}

{{{V}}}{{{""=""}}}{{{expr(4/3)pi*3^3}}}

{{{V}}}{{{""=""}}}{{{36pi}}} cubic centimeters.

Edwin</pre>