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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

1767.0. "Math Magazine Problem 1427" by RUSURE::EDP (Always mount a scratch monkey.) Mon Jun 14 1993 17:17

    Proposed by Gary L. Van Velsir, Anne Arundel Community College, Arnold,
    Maryland.
    
    A bi-fold closet door consists of two one-foot-wide panels, hinged at
    point P.  One of the panels is fixed at the point O (see figure). 
    Assume that as the endpoint Q moves to the right, the door rubs against
    a thick carpet.  What shape will be swept out on the carpet?
    
    [I can't draw the figure, but Q moves to the right in the horizontal
    line containing O, from which Q starts.]

T.R	Title	User	Personal Name	Date	Lines
1767.1		SSAG::LARY	Laughter & hope & a sock in the eye	`Tue Jun 15 1993 05:40`	25
	The outline of the curve sept by the door is the curve f(x) = max(g(x,y), x <= y <= 2), where g(x,y) is the height of the door at abscissa x when points O and Q are separated by y. This can be "simplified" to f(x) = max(g(x,y), x <= y <= min(2x,2)) since no point on the "fixed" half of the door can ever be higher at any given x value than the hinge was at that x value. g(x,y) = 2 * (1 - x/y) * sqrt(1 - y�/4) To find the y that maximizes g(x,y) take the derivative with respect to y, dg(x,y)/dy = 2(x/y�)sqrt(1 - y�/4) + 2(1 - x/y)(-y/4)/sqrt(1-y�/4) Set it equal to zero and solve for y, y = cuberoot(4x) which is between x and 2x for x in [sqrt(2)/2, 2], and in this interval f(x) = g(x,(4x)^1/3) = 2 * (1 - cuberoot(2x�)/2)^3/2 sqrt(2)/2 <= x <= 2 In the interval [0, sqrt(2)/2] the max of g(x,y) is at y = 2x, and f(x) = g(x,2x) = sqrt(1-x�) 0 <= x <= sqrt(2)/2
1767.2		AUSSIE::GARSON	nouveau pauvre	`Fri Jul 02 1993 23:24`	7
	re .1 Checks out. FWIW, the slope is continuous at the point where the two pieces of the answer join. I wasn't able to verify the answer experimentally. My carpet is not shaggy enough.
1767.3		RUSURE::EDP	Always mount a scratch monkey.	`Mon Jun 06 1994 14:26`	9
	The solution published differs from .1 in 1/sqrt(2) <= x <= 2. -- edp Public key fingerprint: 8e ad 63 61 ba 0c 26 86 32 0a 7d 28 db e7 6f 75 To get PGP, FTP /pub/unix/security/crypt/pgp23A.zip from ftp.funet.fi. For FTP access, mail "help" message to DECWRL::FTPmail or open Upsar::Gateways.
1767.4		RUSURE::EDP	Always mount a scratch monkey.	`Wed Jun 15 1994 15:54`	10
	According to the published solution, the path obeys x^(2/3) + y^(2/3) = 2^(2/3) for 1/sqrt(2) <= x <= 2. -- edp Public key fingerprint: 8e ad 63 61 ba 0c 26 86 32 0a 7d 28 db e7 6f 75 To get PGP, FTP /pub/unix/security/crypt/pgp23A.zip from ftp.funet.fi. For FTP access, mail "help" message to DECWRL::FTPmail or open Upsar::Gateways.