Difference between revisions of "2023 AIME II Problems/Problem 14"
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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> | ||
+ | ==Diagram== | ||
+ | <asy> | ||
+ | //Made by Djmathman | ||
+ | size(250); | ||
+ | defaultpen(linewidth(0.6)); | ||
+ | pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y; | ||
+ | pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W; | ||
+ | pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8); | ||
+ | filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2)); | ||
+ | fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9)); | ||
+ | draw(A--B--Z--X--A--Y--C--X^^C--D--Z); | ||
+ | draw(P1--P2--P3--P4--cycle^^D--P4); | ||
+ | dot("$A$",A,S); | ||
+ | dot("$B$",B,S); | ||
+ | dot("$C$",C,N); | ||
+ | dot("$D$",D,N); | ||
+ | label("$\mathcal P$",(-13,4.5)); | ||
+ | </asy> | ||
==Solution (3-d vector analysis, analytic geometry + Calculus)== | ==Solution (3-d vector analysis, analytic geometry + Calculus)== |
Revision as of 14:50, 19 February 2023
Contents
[hide]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
Diagram
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.