Difference between revisions of "2012 AIME I Problems/Problem 8"
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== Solution == | == Solution == | ||
+ | Define a coordinate system with <math>D</math> at the origin and <math>C,</math> <math>A,</math> and <math>H</math> on the <math>x</math>, <math>y</math>, and <math>z</math> axes respectively. The <math>D=(0,0,0),</math> <math>M=(.5,1,0),</math> and <math>N=(1,0,.5).</math> It follows that the plane going through <math>D,</math> <math>M,</math> and <math>N</math> has equation <math>2x-y-4z=0.</math> Let <math>Q = (1,1,.25)</math> be the intersection of this plane and edge <math>BF</math> and let <math>P = (1,2,0).</math> Now since <math>2 \cdot 2 - 2 - 4 \cdot 0 = 0,</math> <math>P</math> is on the plane. Also, <math>P</math> lies on the extensions of segments <math>DM,</math> <math>NQ,</math> and <math>CB</math> so the solid <math>DPCN = DMBCQN + MPBQ</math> is a right triangular pyramid. Note also that pyramid <math>MPBQ</math> is similar to <math>DPCN</math> with scale factor <math>.5</math> and thus the volume of solid <math>DMBCQN,</math> which is one of the solids bounded by the cube and the plane, is <math>[DPCN] - [MPBQ] = \frac{7}{8}[DPCN].</math> But the volume of <math>DPCN</math> is simply the volume of a pyramid with base <math>1</math> and height <math>.5</math> which is <math>\frac{1}{3} \cdot 1 \cdot .5 = \frac{1}{6}.</math> So <math>[DMBCQN] = \frac{7}{8}[DPCN] = \frac{7}{8} \cdot \frac{1}{6} = \frac{7}{48}.</math> Note, however, that this volume is less than <math>.5</math> and thus this solid is the smaller of the two solids. The desired volume is then <math>[ABCDEFGH] - [DMBCQN] = 1 - \frac{7}{48} = \frac{41}{48} \rightarrow p+q = \boxed{089.}</math> | ||
== See also == | == See also == | ||
{{AIME box|year=2012|n=I|num-b=7|num-a=9}} | {{AIME box|year=2012|n=I|num-b=7|num-a=9}} |
Revision as of 18:07, 17 March 2012
Problem 8
Cube labeled as shown below, has edge length and is cut by a planing passing through vertex and the midpoints and of and respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form where and are relatively prime positive integers. Find
Solution
Define a coordinate system with at the origin and and on the , , and axes respectively. The and It follows that the plane going through and has equation Let be the intersection of this plane and edge and let Now since is on the plane. Also, lies on the extensions of segments and so the solid is a right triangular pyramid. Note also that pyramid is similar to with scale factor and thus the volume of solid which is one of the solids bounded by the cube and the plane, is But the volume of is simply the volume of a pyramid with base and height which is So Note, however, that this volume is less than and thus this solid is the smaller of the two solids. The desired volume is then
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |