Difference between revisions of "2023 AIME II Problems/Problem 14"
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Revision as of 19:39, 16 February 2023
==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 .