Difference between revisions of "1996 AIME Problems/Problem 4"

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[[Category:Intermediate Geometry Problems]]
 
[[Category:Intermediate Geometry Problems]]
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Revision as of 19:31, 4 July 2013

Problem

A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x$.

Solution

import three;
size(250);defaultpen(0.7+fontsize(9));
real unit = 0.5; 
real r = 2.8;
triple O=(0,0,0), P=(0,0,unit+unit/(r-1)); dot(P);
draw(O--P); draw(O--(unit,0,0)--(unit,0,unit)--(0,0,unit)); draw(O--(0,unit,0)--(0,unit,unit)--(0,0,unit));
draw((unit,0,0)--(unit,unit,0)--(unit,unit,unit)--(unit,0,unit)); draw((0,unit,0)--(unit,unit,0)--(unit,unit,unit)--(0,unit,unit)); 
draw(P--(r*unit,0,0)--(r*unit,r*unit,0)--(0,r*unit,0)--P); draw(P--(r*unit,r*unit,0)); draw((r*unit,0,0)--(0,0,0)--(0,r*unit,0));
draw(P--(0,0,unit)--(unit,0,unit)--(unit,0,0)--(r*unit,0,0)--P,dashed+blue+linewidth(0.8));
draw(rightanglemark(P,(0,0,unit),(unit,0,unit),1.7));draw(rightanglemark((unit,0,unit),(unit,0,0),(r*unit,0,0),1.7));
label("$x$",(0,0,unit+unit/(r-1)/2),WSW);
label("$1$",(unit/2,0,unit),N);
label("$1$",(unit,0,unit/2),W);
label("$1$",(unit/2,0,0),N);
label("$6$",(unit*(r+1)/2,0,0),N);
label("$7$",(unit*r,unit*r/2,0),SW);
 (Error compiling LaTeX. draw(rightanglemark(P,(0,0,unit),(unit,0,unit),1.7));draw(rightanglemark((unit,0,unit),(unit,0,0),(r*unit,0,0),1.7));
                   ^
112982610b648ae4a5b643e01bc8b68a91f453c1.asy: 12.20: cannot call
'path rightanglemark(pair A, pair B, pair C, real s=<default>)'
with parameters
'(triple, triple, triple, real)')

(Figure not to scale) The area of the square shadow base is $48 + 1 = 49$, and so the sides of the shadow are $7$. Using the similar triangles in blue, $\frac {x}{1} = \frac {1}{6}$, and $\left\lfloor 1000x \right\rfloor = \boxed{166}$.

See also

1996 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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