Difference between revisions of "2023 AIME II Problems/Problem 14"
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==Problem== | ==Problem== | ||
A cube-shaped container has vertices <math>A,</math> <math>B,</math> <math>C,</math> and <math>D,</math> where <math>\overline{AB}</math> and <math>\overline{CD}</math> are parallel edges of the cube, and <math>\overline{AC}</math> and <math>\overline{BD}</math> are diagonals of faces of the cube, as shown. Vertex <math>A</math> of the cube is set on a horizontal plane <math>\mathcal{P}</math> so that the plane of the rectangle <math>ABDC</math> is perpendicular to <math>\mathcal{P},</math> vertex <math>B</math> is <math>2</math> meters above <math>\mathcal{P},</math> vertex <math>C</math> is <math>8</math> meters above <math>\mathcal{P},</math> and vertex <math>D</math> is <math>10</math> meters above <math>\mathcal{P}.</math> The cube contains water whose surface is parallel to <math>\mathcal{P}</math> at a height of <math>7</math> meters above <math>\mathcal{P}.</math> The volume of water is <math>\frac{m}{n}</math> cubic meters, where <math>m</math> and <math>n</math> are relatively prime positive intgers. Find <math>m+n.</math> | A cube-shaped container has vertices <math>A,</math> <math>B,</math> <math>C,</math> and <math>D,</math> where <math>\overline{AB}</math> and <math>\overline{CD}</math> are parallel edges of the cube, and <math>\overline{AC}</math> and <math>\overline{BD}</math> are diagonals of faces of the cube, as shown. Vertex <math>A</math> of the cube is set on a horizontal plane <math>\mathcal{P}</math> so that the plane of the rectangle <math>ABDC</math> is perpendicular to <math>\mathcal{P},</math> vertex <math>B</math> is <math>2</math> meters above <math>\mathcal{P},</math> vertex <math>C</math> is <math>8</math> meters above <math>\mathcal{P},</math> and vertex <math>D</math> is <math>10</math> meters above <math>\mathcal{P}.</math> The cube contains water whose surface is parallel to <math>\mathcal{P}</math> at a height of <math>7</math> meters above <math>\mathcal{P}.</math> The volume of water is <math>\frac{m}{n}</math> cubic meters, where <math>m</math> and <math>n</math> are relatively prime positive intgers. Find <math>m+n.</math> | ||
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==Solution (3-d vector analysis, analytic geometry + Calculus)== | ==Solution (3-d vector analysis, analytic geometry + Calculus)== |
Revision as of 16:21, 18 February 2023
Problem
A cube-shaped container has vertices
and
where
and
are parallel edges of the cube, and
and
are diagonals of faces of the cube, as shown. Vertex
of the cube is set on a horizontal plane
so that the plane of the rectangle
is perpendicular to
vertex
is
meters above
vertex
is
meters above
and vertex
is
meters above
The cube contains water whose surface is parallel to
at a height of
meters above
The volume of water is
cubic meters, where
and
are relatively prime positive intgers. Find
Solution (3-d vector analysis, analytic geometry + Calculus)
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)
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.