Question 1114815
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I read and interpret the condition by different way: two spherical solids are taken out by cutting from a solid cube
in a way that their centers are located on the &nbsp;3D &nbsp;(=longest) diagonal of the cube.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(This condition provides the maximum radius and maximum volume to each of the two spheres).



Then it is clear that these spheres touch each other at the middle of the &nbsp;3D&nbsp; diagonal of the cube.


The length of the longest 3D diagonal of this cube is  &nbsp;{{{13*sqrt(3)}}} cm.


If "r" is the radius of the sphere, then  &nbsp;&nbsp;{{{r*sqrt(3)+r}}} = {{{6.5*sqrt(3)}}} cm.


Hence,  &nbsp;&nbsp;r = {{{(6.5*sqrt(3))/(1 + sqrt(3))}}} = 4.12 cm.


Then the volume of each sphere  &nbsp;&nbsp;V = {{{(4/3)*pi*r^3}}} = {{{(4/3)*3.14*4.12^3}}} = 292.8 cm^3.
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Solved.