Difference between revisions of "1996 AIME Problems/Problem 4"
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== Problem == | == Problem == | ||
| − | A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is <math>x</math> 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 <math>1000x</math>. | + | A wooden [[cube]], whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is <math>x</math> 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 <math>1000x</math>. |
== Solution == | == Solution == | ||
| − | {{ | + | <center><asy> |
| + | 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); | ||
| + | </asy></center> | ||
| + | (Figure not to scale) The area of the square shadow base is <math>48 + 1 = 49</math>, and so the sides of the shadow are <math>7</math>. Using the similar triangles in blue, | ||
| + | <math>\frac {x}{1} = \frac {1}{6}</math>, and <math>\left\lfloor 1000x \right\rfloor = \boxed{166}</math>. | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=1996|num-b=3|num-a=5}} | |
| − | + | [[Category:Intermediate Geometry Problems]] | |
Revision as of 19:24, 28 March 2008
Problem
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is 
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 
.
Solution
![[asy] 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); [/asy]](http://latex.artofproblemsolving.com/9/5/c/95ce6725473f91467d41a0e35cee50288617958d.png)
(Figure not to scale) The area of the square shadow base is 
, and so the sides of the shadow are 
. Using the similar triangles in blue,
, and 
.
See also
| 1996 AIME (Problems • Answer Key • Resources) | ||
| 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 | ||
