310 helix - Revision history

en>Mithoron: minor

2014-11-14T13:06:26Z

minor

← Older revision		Revision as of 15:06, 14 November 2014
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	Hi there~~, I am Andrew Berryhill~~. ~~Invoicing~~ is my ~~occupation~~. ~~One~~ of ~~the things~~ she enjoys ~~most~~ is ~~canoeing~~ and she~~'s been doing~~ it ~~for quite~~ a ~~while~~. ~~Kentucky~~ is ~~where I've always been living~~.<br><br>my ~~web site~~: ~~free psychic (~~[http://~~www~~.~~octionx~~.~~sinfauganda~~.co.~~ug/node/22469 blog~~])		Hi there. My name is Garland even though it is not the title on my beginning certification. Kansas is exactly where her house is but she requirements to transfer because of her family members. What she enjoys performing is bottle tops gathering and she is attempting to make it a occupation. Bookkeeping is what he does.<br><br>Also visit my blog post: [http://Www.nanumgdc.com/forest/board_xUbo69/841259 Www.nanumgdc.com]

130.236.213.1 at 08:01, 25 February 2014

2014-02-25T08:01:21Z

← Older revision		Revision as of 10:01, 25 February 2014
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	~~The '''crossed ladders problem''' is a [[puzzle]] of unknown origin that has appeared in various publications and regularly reappears in Web pages and [[Usenet]] discussions.~~		Hi there, I am Andrew Berryhill. Invoicing is my occupation. One of the things she enjoys most is canoeing and she's been doing it for quite a while. Kentucky is where I've always been living.<br><br>my web site: free psychic ([http://www.octionx.sinfauganda.co.ug/node/22469 blog])

	~~==The problem==~~

	~~[[File:CrossedLadders~~.~~svg\|thumb\|right\|Crossed ladders. ''h''~~ is ~~half the harmonic mean of ''A'' and ''B'']]~~

	~~Two ladders of lengths ''a'' and ''b'' lie oppositely across an alley, as shown in the figure~~. ~~The ladders cross at a height~~ of ~~''h'' above~~ the ~~alley floor. What~~ is ~~the width of the alley?~~

	~~Martin Gardner presents and discusses the problem in his book of mathematical puzzles published in 1979~~ and cites references to it as early as 1895. The Crossed Ladders Problem may appear in various forms, with variations in name, using various lengths and heights, or requesting unusual solutions such as cases where all values are integers. Its charm has been attributed to a seeming simplicity which can quickly devolve into an "algebraic mess" [characterization attributed by Gardner to D. F. Church].

	~~== Solution summary ==~~

	The problem may be reduced to the quartic equation ''x''<sup> 3</sup>(''x'' − ''c'') − 1 = 0, which can be solved by approximation methods, as suggested by Gardner, or the quartic may be solved in [[closed-form expression\|closed form]] by [[Quartic equation#Summary of Ferrari's method\|Ferrari's method]]. Once ''x'' is obtained, the width of the alley is readily calculated. A derivation of the quartic is outlined below. Note the potentially confusing fact that the requested unknown, ''w'', does not appear.

	~~: Eq 1: Divide the baseline into two parts at the point where~~ it ~~meets <math> h \,</math>, and call the left and right parts <math> w_1 \,</math> and <math>w_2 \,</math>, respectively~~. ~~The angle where <math> a \,</math> meets <math> w \,</math>~~ is ~~common to two similar triangles with bases <math>w\,</math> and <math>w_1\,</math> respectively. The angle~~ where ~~<math> b \,</math> meets <math> w \,</math> is common to two similar triangles with bases <math>w\,</math> and <math>w_2\,</math> respectively~~. ~~This tells us that~~

	: <~~math~~> ~~\frac{B}{w} = \frac{h}{w_1} \,~~ <~~/math~~> ~~and <math> \frac{A}{w} = \frac{h}{w_2} \, </math>~~

	: ~~which we can then re-arrange~~ (~~using <math>w_1 + w_2 = w</math>) to get~~

	~~: <math>\frac{1}{A} + \frac{1}{B} = \frac{1}{h} \,</math>~~

	~~: from which we get~~

	~~: <math>A B = h (A + B) \, </math>.~~


	~~: Eq 2: Using the~~ [~~[Pythagorean theorem]], we can see that~~
	: ~~<math>w^2+B^2=a^2<~~/~~math> and <math>w^2+A^2=b^2<~~/~~math>~~.
	~~: By isolating w² on both equations, we see that~~
	~~: <math>a^2-B^2=b^2-A^2</math>~~
	:
	~~: which can be rearranged and factored into~~
	~~: <math>a^2 - b^2 = (B + A) (B - A) \,</math>~~.


	~~: Eq 3: Square (Eq 2) and combine with (Eq 1)~~
	~~: <math>(a^2 - b^2)^2 = (A+B)^3(A+B-4h) \,</math>~~

	~~Let~~

	~~: <math> x = \frac{A + B}{\sqrt{a^2 - b^2}}, \,</math>~~
	~~: <math> c = \frac{4 h}{\sqrt{a^2-b^2}}~~. ~~\,</math>~~

	~~Then~~

	~~: <math> x^3(x-c)-1 = 0~~.~~\,<~~/~~math> (same as Eq 3)~~

	~~The above fourth power equation can be solved for ''x'' using any available method. The width of the alley is then found by using the value found for ''x''.~~

	A quartic equation has four solutions, and only one solution for this equation matches the problem as presented. Another solution is for a case where one ladder (and wall) is below ground level and the other above ground level. In this case the ladders do not actually cross, but the intersection of their extensions do so at the specified height. The other two solutions are a pair of conjugate complex numbers. The equation does not have the ladder lengths explicitly defined, only the difference of their squares, so one could take the length as any value that makes them cross, and the wall spacing would be defined as between where the ladders intersect the walls.

	Surprisingly as the wall spacing approaches zero, the height of the crossing approaches <math>w = \frac{ab}{a+b}</math>. As the solutions to the equation involve square roots, negative roots are equally valid so both ladders and walls can be below ground level and with them in opposing sense, they can be interchanged.

	~~==References==~~
	* Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 62–64, 1979.

	~~==External links==~~
	* {{MathWorld \|title=Crossed Ladders Theorem \|urlname=CrossedLaddersTheorem}}
	* [http://demonstrations.wolfram.com/~~CrossedLaddersTheorem/ Crossed Ladders Theorem] by Jay Warendorff, the [[Wolfram Demonstrations Project]~~].
	* [http://users.softlab.ntua.gr/~ttsiod/ladders.html Solving the crossing ladders puzzle (with Python, GNU GSL, Octave, Maxima and Sage)].

	~~[[Category:Puzzles]]~~
	~~[[Category:Elementary geometry]]~~

en>EmausBot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q229812

2013-04-15T16:10:55Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q229812

← Older revision		Revision as of 18:10, 15 April 2013
Line 1:		Line 1:
	The ~~author~~'~~s name~~ is ~~Christy Brookins~~. ~~Invoicing~~ is ~~my profession~~. ~~One~~ of the ~~things she enjoys most~~ is ~~canoeing~~ and ~~she~~'s ~~been doing~~ it ~~for quite~~ a ~~whilst~~. ~~Her family members lives in Alaska but her spouse wants them~~ to ~~move~~.<br><br>~~Feel free~~ to ~~surf to my webpage~~ - ~~phone psychic~~ (~~[https~~://~~www~~.~~machlitim~~.~~org~~.il/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~12300 Get~~ the ~~facts~~])		The '''crossed ladders problem''' is a [[puzzle]] of unknown origin that has appeared in various publications and regularly reappears in Web pages and [[Usenet]] discussions.

			==The problem==

			[[File:CrossedLadders.svg\|thumb\|right\|Crossed ladders. ''h'' is half the harmonic mean of ''A'' and ''B'']]

			Two ladders of lengths ''a'' and ''b'' lie oppositely across an alley, as shown in the figure. The ladders cross at a height of ''h'' above the alley floor. What is the width of the alley?

			Martin Gardner presents and discusses the problem in his book of mathematical puzzles published in 1979 and cites references to it as early as 1895. The Crossed Ladders Problem may appear in various forms, with variations in name, using various lengths and heights, or requesting unusual solutions such as cases where all values are integers. Its charm has been attributed to a seeming simplicity which can quickly devolve into an "algebraic mess" [characterization attributed by Gardner to D. F. Church].

			== Solution summary ==

			The problem may be reduced to the quartic equation ''x''<sup> 3</sup>(''x'' − ''c'') − 1 = 0, which can be solved by approximation methods, as suggested by Gardner, or the quartic may be solved in [[closed-form expression\|closed form]] by [[Quartic equation#Summary of Ferrari's method\|Ferrari's method]]. Once ''x'' is obtained, the width of the alley is readily calculated. A derivation of the quartic is outlined below. Note the potentially confusing fact that the requested unknown, ''w'', does not appear.

			: Eq 1: Divide the baseline into two parts at the point where it meets <math> h \,</math>, and call the left and right parts <math> w_1 \,</math> and <math>w_2 \,</math>, respectively. The angle where <math> a \,</math> meets <math> w \,</math> is common to two similar triangles with bases <math>w\,</math> and <math>w_1\,</math> respectively. The angle where <math> b \,</math> meets <math> w \,</math> is common to two similar triangles with bases <math>w\,</math> and <math>w_2\,</math> respectively. This tells us that

			: <math> \frac{B}{w} = \frac{h}{w_1} \, </math> and <math> \frac{A}{w} = \frac{h}{w_2} \, </math>

			: which we can then re-arrange (using <math>w_1 + w_2 = w</math>) to get

			: <math>\frac{1}{A} + \frac{1}{B} = \frac{1}{h} \,</math>

			: from which we get

			: <math>A B = h (A + B) \, </math>.


			: Eq 2: Using the [[Pythagorean theorem]], we can see that
			: <math>w^2+B^2=a^2</math> and <math>w^2+A^2=b^2</math>.
			: By isolating w² on both equations, we see that
			: <math>a^2-B^2=b^2-A^2</math>
			:
			: which can be rearranged and factored into
			: <math>a^2 - b^2 = (B + A) (B - A) \,</math>.


			: Eq 3: Square (Eq 2) and combine with (Eq 1)
			: <math>(a^2 - b^2)^2 = (A+B)^3(A+B-4h) \,</math>

			Let

			: <math> x = \frac{A + B}{\sqrt{a^2 - b^2}}, \,</math>
			: <math> c = \frac{4 h}{\sqrt{a^2-b^2}}. \,</math>

			Then

			: <math> x^3(x-c)-1 = 0.\,</math> (same as Eq 3)

			The above fourth power equation can be solved for ''x'' using any available method. The width of the alley is then found by using the value found for ''x''.

			A quartic equation has four solutions, and only one solution for this equation matches the problem as presented. Another solution is for a case where one ladder (and wall) is below ground level and the other above ground level. In this case the ladders do not actually cross, but the intersection of their extensions do so at the specified height. The other two solutions are a pair of conjugate complex numbers. The equation does not have the ladder lengths explicitly defined, only the difference of their squares, so one could take the length as any value that makes them cross, and the wall spacing would be defined as between where the ladders intersect the walls.

			Surprisingly as the wall spacing approaches zero, the height of the crossing approaches <math>w = \frac{ab}{a+b}</math>. As the solutions to the equation involve square roots, negative roots are equally valid so both ladders and walls can be below ground level and with them in opposing sense, they can be interchanged.

			==References==
			* Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 62–64, 1979.

			==External links==
			* {{MathWorld \|title=Crossed Ladders Theorem \|urlname=CrossedLaddersTheorem}}
			* [http://demonstrations.wolfram.com/CrossedLaddersTheorem/ Crossed Ladders Theorem] by Jay Warendorff, the [[Wolfram Demonstrations Project]].
			* [http://users.softlab.ntua.gr/~ttsiod/ladders.html Solving the crossing ladders puzzle (with Python, GNU GSL, Octave, Maxima and Sage)].

			[[Category:Puzzles]]
			[[Category:Elementary geometry]]

en>Redfiona99: Removing repetition

2012-02-16T17:54:39Z

Removing repetition

New page

The author's name is Christy Brookins. Invoicing is my profession. One of the things she enjoys most is canoeing and she's been doing it for quite a whilst. Her family members lives in Alaska but her spouse wants them to move.<br><br>Feel free to surf to my webpage - phone psychic ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 Get the facts])