AIME 2015b Problem 9

A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$. [asy] import three; import solids; size(5cm); currentprojection=orthographic(1,-1/6,1/6); draw(surface(revolution((0,0,0),(-2,-2sqrt(3),0)--(-2,-2sqrt(3),-10),Z,0,360)),white,nolight); triple A =(8sqrt(6)/3,0,8sqrt(3)/3), B = (-4sqrt(6)/3,4sqrt(2),8sqrt(3)/3), C = (-4sqrt(6)/3,-4sqrt(2),8sqrt(3)/3), X = (0,0,-2sqrt(2)); draw(X--X+A--X+A+B--X+A+B+C); draw(X--X+B--X+A+B); draw(X--X+C--X+A+C--X+A+B+C); draw(X+A--X+A+C); draw(X+C--X+C+B--X+A+B+C,linetype("2 4")); draw(X+B--X+C+B,linetype("2 4")); draw(surface(revolution((0,0,0),(-2,-2sqrt(3),0)--(-2,-2sqrt(3),-10),Z,0,240)),white,nolight); draw((-2,-2sqrt(3),0)..(4,0,0)..(-2,2sqrt(3),0)); draw((-4cos(atan(5)),-4sin(atan(5)),0)--(-4cos(atan(5)),-4sin(atan(5)),-10)..(4,0,-10)..(4cos(atan(5)),4sin(atan(5)),-10)--(4cos(atan(5)),4sin(atan(5)),0)); draw((-2,-2sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype("2 4")); [/asy]