Longest Ladder: > > | A | > |<----->| > | | > |\ | > | \ | > | \ | > | \ | > | \ | > | \ | > | \`----------- > | \ ^ > | \ |B > | theta\ v > `------------------- > > Given corner of hallways of widths A and B as shown in the diagram, what > is the longest ladder which can be moved around the corner? SPOILER If the angle between the ladder and the wall is theta (as shown above), then the length of the ladder is length(theta) = A/cos(theta) + B/sin(theta). Minimize length = A/cos(theta) + B/sin(theta) d(length)/d(theta) = A*sin(theta)/cos(theta)^2 - B*cos(theta)/sin(theta)^2 (?) = 0 A*sin(theta)/cos(theta)^2 = B*cos(theta)/sin(theta)^2 B/A = sin(theta)^3/cos(theta()^3 = tan(theta)^3 If you use the trigonometric formulas cos^2 t = 1/(1 + tan^2 t) and sin t = tan t cos t, and plug through the algebra, that the formula for the length reduces to length = A/cos(t) + B/sin(t) = (A + B/tan(t))/cos(t) = (A + B/tan(t))(1 + tan^2(t))^(1/2) = (A + B^(2/3) A^(1/3)) (1 + B^(2/3)/A^(2/3))^(1/2) = (A^(2/3) + B^(2/3))^(3/2) ---------------------------------------------------------------------- From X Wed Jun 30 12:24:49 MET DST 1993 Article: 4695 of rec.puzzles Subject: Hallway and ladder without calculus From: kubo@zariski.harvard.edu (Tal Kubo) Several calculus-based solutions have been posted. There are at least two ways that don't use calculus: 1. H\"older's inequality (ref: Hardy, Littlewood, Polya, "Inequalities") combined with the relation A/x + B/y =1 where x and y are the distances from the corner to the points where the ladder touches the walls, can be used to bound x^2 + y^2 in terms of A and B. This gives a one-line solution. 2. Geometric arguments that the envelope of a ladder of length L whose ends slide along the positive x and y axes of a coordinate plane is (one arch of) the figure traced out by a point on a circle of radius L/4 rolling inside a circle of radius L. The equation of the envelope can be derived from this. For the details, see Vasiliev and Gutenmacher, "Straight Lines and Curves". -- Tal Kubo ---------------------------------------------------------------------- Straight line through P = (A,B). It crosses the axis at (x,0) and (0,y). A/x + B/y = 1 We get a right triangle (x,0), (0,y) and (0,0). Problem 0: minimize L0 = xy (x/A * y/B)^(1/2) >= ( ((x/A)^(-1) + (y/B)^(-1))/2 )^(-1) by geometric-harmonic-mean = 2 We get L0 >= 4AB with equality iff x / y = A / B. Problem 1: minimize L1 = x+y (x + y)^(1/2) = (x + y)^(1/2) * (A/x + B/y)^(1/2) >= A^(1/2) + B^(1/2) by H\"older's inequality. We get L1^(1/2) >= A^(1/2) + B^(1/2) with equality iff x^2 / y^2 = A / B. Problem 2: minimize L2 = (x^2+y^2)^(1/2) (x^2 + y^2)^(1/3) = (x^2 + y^2)^(1/3) * (A/x + B/y)^(2/3) >= A^(2/3) + B^(2/3) by H\"older's inequality. We get L2^(2/3) >= A^(2/3) + B^(2/3) with equality iff x^3 / y^3 = A / B. We used H\"older's inequality in the form: Let p, q > 0 with p + q = 1 and x1, x2, y1, y2 non negative then (x1 + x2)^p (y1 + y2)^q >= x1^p y1^q + x2^p y2^q with equality iff (x1, x2), (y1, y2) are linear dependent. - - - - - - - - - - - - - - - - - - - - - Plane through P = (A,B,C). It crosses the axis at (x,0,0), (0,y,0) and (0,0,z). A/x + B/y + C/z = 1 We get a right tetrahedron (x,0,0), (0,y,0), (0,0,z) and (0,0,0). Problem 0.3: minimize L0 = xyz (x/A * y/B * z/C)^(1/3) >= ( ((x/A)^(-1) + (y/B)^(-1) + (z/C)^(-1))/3 )^(-1) by geometric-harmonic-mean = 3 We get L0 >= 27ABC with equality iff x : y : z = A : B : C. Problem 1.3: minimize L1 = x+y+z (x + y + z)^(1/2) = (x + y + z)^(1/2) * (A/x + B/y + C/z)^(1/2) >= A^(1/2) + B^(1/2) + C^(1/2) by H\"older's inequality. We get L1^(1/2) >= A^(1/2) + B^(1/2) + C^(1/2) with equality iff x^2 : y^2 : z^2 = A : B : C. References: - Raymond T. Boute; Moving a Rectangle around a Corner - Geometrically, American Mathematical Monthly 111 (May 2004) 435-437 - kinematics: the instantaneous center of rotation (ICR) - the ladder problem (using ICR and similar triangles) - the couch problem - Raymond T. Boute; Simple geometric solutions to De l'Hospital's pulley problem, College Mathematics Journal 30 (1999) 311-314 - ratio chaining technique - Victor L. Gutenmacher; Nikolay B. Vasilyev; Lines and Curves, A Practical Geometry Handbook, Birkhaeuser Publication, Boston, 2004 ISBN: 0-817-64161-0 - Nikolay B. Vasilyev; Victor L. Gutenmacher; Straight Lines and Curves, Mir Publishers, Moscow, (1985) - Benno Grabinger; Vom Garagentor zur Astroide, Der Mathematikunterricht 43:2 (1997) 23-37. - envelope - Bettina Helms, Claudia Röttgen, Julia Dierker; Passt das Auto in die Garage? Jülich, 2003 http://www.mathematik-piechatzek.de/Projekt_Garage/Garagentor/index.htm - G.H. Hardy, J.E. Littlewood, and G. Pólya; Inequalities, Cambridge Univ. Press, Cambridge, 1934. Zbl 0010.107 - Ludwig Kiepert; Grundriß der Differential-Rechnung, 1. Band: Funktionen von einer unabhängigen Veränderlichen Hannover, Helwingsche Verlagsbuchhandlung, 1921, 14te Auflage - Section 65: Verschiedene Aufgaben aus der Theorie der Maxima und Minima Aufgabe 17. p337-338 Wie hoch muss die Tür eines Turmes mit der Breite gleich a mindestens sein, damit man eine Leiter von der Länge gleich l in den Turm hineinbringen kann? - N. Miller; The problem of a non-vanishing girder rounding a corner, American Mathematical Monthly 56 (1949) 177-179 - calculus with trigonometry - C. Moretti; Moving a couch around a corner, College Mathematics Journal 33 (2002) 196-200 - calculus - Karl Schuler; Maxima-Minima-Aufgaben, Archimedes, Sonderheft 1966 - gives Schlömilch's (1858) proof of the AM-GM-Inequality. - plane through a point, cutting of a right tetrahedron - Hans Schupp, Heinz Dabrock; Höhere Kurven, Situative, mathematische, historische und didaktische Aspekte, Lehrbücher und Monographien zur Didaktik der Mathematik 28, BI Wissenschaftsverlag, Mannheim, 1995, ISBN 3-411-17221-5 - Chapter 1: Kurven rund um die Gleitstrecke. p1-14 - Hans Schupp; Optimieren, Extremwertbestimmung im Mathematikunterricht, Lehrbücher und Monographien zur Didaktik der Mathematik 20, BI Wissenschaftsverlag, Mannheim, 1992, ISBN 3-411-15771-2 - Chapter 4.1: Methoden - Chapter 4.1.6: Infinitesimalkalkül 3 Beispiele: Gerade durch gegebenen Punkt (a,b) gesucht. p82-84 Beispiel1: Minimiere Flaeche Beispiel2: Minimiere Hypothenuse (Leiter um die Ecke Problem) Beispiel3: Minimiere Umfang - Eckard Specht; geometria - scientiae atlantis. 300+ Aufgaben zur Geometrie und zu Ungleichungen insbesondere zur Vorbereitung auf Mathematik-Olympiaden, Otto-von-Guericke-Universität Magdeburg, 2001, ISBN 3-929757-39-7 - Kapitel U: Ungleichungen - Eckard Specht; 470+ Mathematik-Aufgaben zum Training in Vorbereitung auf Olympiaden und Wettbewerbe http://www.math4u.de/ (online version) - Kapitel U: Ungleichungen Problem U8: (Verallgemeinerte) Höldersche Ungleichungen References Astroid: - Robert FERRÉOL, Jacques MANDONNET; ASTROÏDE, http://www.mathcurve.com/courbes2d/astroid/astroid.shtml - Eric W. Weisstein; "Astroid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Astroid.html - Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, 1967 - Chapter 6: The Astroid. p52-61 - Hans-Jürgen Caspar; Mathematik: Die Leiter und die Sternkurve, http://matheplanet.com/default3.html?call=article.php?sid=489 - Xah Lee; Astroid, http://xahlee.org/SpecialPlaneCurves_dir/Astroid_dir/astroid.html Web Links: - Marek Szapiel; The Longest Ladder, http://archives.math.utk.edu/visual.calculus/3/applications.2/ Shortest Ladder over a Fence, http://archives.math.utk.edu/visual.calculus/3/applications.4/ - matheplanet; Malerarbeit mit Hindernissen, http://matheplanet.com/matheplanet/nuke/html/viewtopic.php?topic=32772 -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/