A088896 - OEIS

%I #10 Mar 31 2012 10:26:02

%S 125,1000,2197,3375,4913,8000,15625,17576,24389,27000,39304,42875,

%T 50653,59319,64000,68921,91125,125000,132651,140608,148877,166375,

%U 195112,216000,226981,274625,314432,343000,389017,405224,421875,474552,512000

%N Length of longest integral ladder that can be moved horizontally around the right angled corner where two hallway corridors of integral widths meet.

%C The set of values for the integral-widths corridors and longest ladder are merely the cubes of Pythagorean triples, viz. (A046083, A046084, A009000).

%C The corridors' widths may be parametrically expressed as d*(sin x)^3 and d*(cos x)^3, for a longest ladder length d making an angle x with one of the corridors.

%C A given ladder, however, is maximum-corner-bending for a family of infinite pairs of perpendicular corridor widths and that the envelope of the maximum bending positions is that of a sliding rod against the outer wall, which is a branch of an astroid or four-cusped hypocycloid.

%D E. Mendelson, 3000 Solved Problems in Calculus, Chapter 16 Problem 16.56 pp. 131, Mc Graw-Hill 1988.

%D M. Spiegel, Theory and Problems of Advanced Calculus, Chapter 4 Problem 40 pp. 75, Mc Graw-Hill 1974.

%H C. Azeredo, <a href="http://www.mtm.ufsc.br/~azeredo/calculos/Acalculo/x/aplicderiv/ladder.html">The Ladder Problem</a>

%H L. Husch and M. Szapiel, <a href="http://archives.math.utk.edu/visual.calculus/3/applications.2">The Longest Ladder</a>

%H M. Kantor, Knox College, <a href="http://math.knox.edu/puzzles/Catalog-Old/current_puzzle.html">Puzzle of the Week</a>

%H J. J. O'Connor and E. R. Robertson, <a href="http://www-groups.dcs.st-and.ac.uk/~history/Curves/Astroid.html">Astroid</a>

%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/ladder">longest ladder</a>

%H D. Sjerve, <a href="http://www.math.ubc.ca/~sjer/math100sec101/sols7.pdf">Solution to problem No.3</a>

%H W. H. Steeb, <a href="http://issc.rau.ac.za/appliedmaths/tgw3a/png/tgw3a6s.html">Solved Problem</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Astroid.html">Astroid</a>

%F a(n)=d^3, where d=A009003(n).

%K nonn

%O 1,1

%A _Lekraj Beedassy_, Nov 28 2003