2023 AIME II Problems/Problem 14
Solution (3-d vector analysis, analytic geometry)
We introduce a Cartesian coordinate system to the diagram.
We put the origin at . We let the
-components of
,
,
be positive.
We set the
-axis in a direction such that
is on the
plane.
The coordinates of ,
,
are
,
,
.
Because ,
.
Thus,
Because is a diagonal of a face,
.
Thus,
Because plane is perpendicular to plan
,
.
Thus,
Jointly solving (1), (2), (3), we get one solution ,
,
.
Thus, the side length of the cube is
.
Denote by and
two vertices such that
and
are two edges, and satisfy the right-hand rule that
.
Now, we compute the coordinates of
and
.
Because , we have
,
,
.
Hence,
By solving these equations, we get \[ y_P^2 + y_Q^2 = 36 . \]
In addition, we have .
Thus,
,
.
Therefore, the volume of the water is
Define ,
,
.
Thus,
Define .
Thus,
Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)