Question 939782
{{{x<y}}}


{{{system(x^2+x^2+x^2=d^2,y^2+y^2+y^2=D^2)}}}



{{{system(d=sqrt(3x^2),D=sqrt(3y^2))}}}


{{{system(x*sqrt(3)=d,y*sqrt(3)=D)}}}


{{{x=d/sqrt(3)}}} and {{{y=D/sqrt(3)}}}


{{{y/x=D/d}}}



Look specifically at the larger cube seemingly diagonal 42 and volume 64.
{{{y*sqrt(3)=D}}}
{{{y=D/sqrt(3)}}}
{{{y=42/sqrt(3)}}}
Rationalize denom;
{{{42sqrt(3)/3}}}
{{{y=14*sqrt(3)}}}


Use the proportion earlier found.
{{{x/y=D/d}}}
{{{x=yD/d}}}
{{{x=14*sqrt(3)*64/22}}}
{{{highlight_green(x=7*64*sqrt(3)/11)}}}----side length of the smaller cube, so raise all this to power 3 to get volume.


{{{v=x^3=122.18181818*sqrt(3)=highlight((122&2/11)sqrt(3))}}}