2015 AIME II Problems/Problem 9
Problem
A cylindrical barrel with radius feet and height
feet is full of water. A solid cube with side length
feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is
cubic feet. Find
.
Solution
Our aim is to find the volume of the part of the cube submerged in the cylinder. In the problem, since three edges emanate from each vertex, the boundary of the cylinder touches the cube at three points. Because the space diagonal of the cube is vertical, by the symmetry of the cube, the three points form an equilateral triangle. Because the radius of the circle is 4, by the Law of Cosines, the side length s of the equilateral triangle is
so . Again by the symmetry of the cube, the volume we want to find is the volume of a tetrahedron with right angles on all faces at the submerged vertex, so since the lengths of the legs of the tetrahedron are $\frac{4\sqrt{3}}{\sqrt{2} = 2\sqrt{6}$ (Error compiling LaTeX. Unknown error_msg) (the three triangular faces touching the submerged vertex are all 45-45-90 trianlges) so
so
.