AIME Problem:

Discus: Individual Schools: US News Top 25: Massachusetts Institute of Technology: 2004 Archive: AIME Problem:
 By 007 (007) on Thursday, July 01, 2004 - 04:56 am: Edit

A circle radius 1 is randomly placed so that it lies entirely inside a 15 x 36 rectangle ABCD. Find the probability that it does not meet the diagonal AC.

how to do it?

 By Devious (Devious) on Thursday, July 01, 2004 - 07:16 am: Edit

The triangle the diagonal makes would have base 36cm and height 15cm. Imagine another triangle inside that one with base 35cm and height 14cm. This leaves a small trapezium between the hypotenuses of both triangles and the 'unused' 1cm of base and height. Also note that inside the rectangle, there'll be a pair of these triangles and trapeziums. So as long as the center of the circle remains outside both trapziums, it won't touch the diagonal. Find the areas of the trapezium and the triangles, then apply whatever method to find the probability.

Or, hmmm... Find the length of the diagonal (39), and the area of the rectangle (540). The probably = 39/540 = 13/180.

I haven't really learned probability, so this is a stab in the dark. Both answers'll probably be completely wrong.

 By Texas137 (Texas137) on Thursday, July 01, 2004 - 09:30 am: Edit

a good place to get answers to AIME problems, if you don't already know about it, is www.artofproblemsolving.com

 By Devious (Devious) on Thursday, July 01, 2004 - 12:11 pm: Edit

I've thought about this a little further, and I think the smaller triangle (in my first attempt) should actually have a 34cm base and a 13cm altitude.

 By Joeperson (Joeperson) on Thursday, July 01, 2004 - 09:00 pm: Edit

tan( 90 - Arctan(15/36) / 2 ) = 5
tan( 90 - Arctan(36/15) / 2 ) = 1.5

Possible area = 13*34 = 442
Area 1 unit from diagonal = (14-1.5)(35-5) = 375
Probability = 375/442 = .8484

But that's probably no help because I can't explain it without pictures.

Basically, find the area at least 1 unit away from the sides of the rectangle and its diagonal and divide that by the area at least 1 unit away from just the sides.

 By Webhappy2 (Webhappy2) on Thursday, July 01, 2004 - 10:47 pm: Edit

Listen to Texas. This is from the most recent AIME, so it's on the AOPS math jam transcripts. I remember seeing their nice diagrams for this problem.

 By 007 (007) on Friday, July 02, 2004 - 03:25 am: Edit

Joeperson: its like you try and find the probablilty of placing the cetre of the circle right?? so you will have a smaller rectangle of 34x13 inside the bigger one. and then the centre can be anywhere provided it is 1cm away from the diagonal??

thanx i think i got it

 By Texas137 (Texas137) on Friday, July 02, 2004 - 11:53 am: Edit

http://www.artofproblemsolving.com