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Difference between revisions of "2016 AIME I Problems/Problem 4"

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==Problem==
 
==Problem==
 
A right prism with height <math>h</math> has bases that are regular hexagons with sides of length 12. A vertex <math>A</math> of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain <math>A</math> measures <math>60</math> degrees. Find <math>h^2</math>.
 
A right prism with height <math>h</math> has bases that are regular hexagons with sides of length 12. A vertex <math>A</math> of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain <math>A</math> measures <math>60</math> degrees. Find <math>h^2</math>.
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== Solution == 
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Let B and C be the vertices adjacent to A on the same base as A and let D be the other vertex of the triangular pyramid.  Then <math>\angle CAB = 120^\circ</math> so <math>[ABC] = \frac{1}{2} \cdot AB \cdot AC \cdot \text{sin}(120^\circ) = 36\sqrt{2}</math>.  Let <math>D</math> be the foot of the altitude from <math>B</math> to <math>AC</math>.

Revision as of 18:06, 4 March 2016

Problem

A right prism with height $h$ has bases that are regular hexagons with sides of length 12. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60$ degrees. Find $h^2$.

Solution

Let B and C be the vertices adjacent to A on the same base as A and let D be the other vertex of the triangular pyramid. Then $\angle CAB = 120^\circ$ so $[ABC] = \frac{1}{2} \cdot AB \cdot AC \cdot \text{sin}(120^\circ) = 36\sqrt{2}$. Let $D$ be the foot of the altitude from $B$ to $AC$.

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