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The foot of the ladder has to rest on the ground, therefore it is constrained to move horizontally only. The body of the ladder passes through a point fixed at 6ft. above the ground. At any time the ladder makes an angle A with the ground. As A varies the position of the top of the ladder varies based on the two imposed restraints. The top of the ladder will sweep out a complex curve which is a function of the length of the ladder, the height of the point, and angle A. Where this curve reaches a turning point is the horizontal position of the wall. The first wall position to be derived must be from the curve generated by the 12ft. ladder. The equation for the curve is fairly simple.

The coordinates for the curve are x=12cosA-6/tanA y=12sinA-6 (12ft. ladder).
The turning point occurs when x is a maximum (angle A=52.004988671281deg.)
The foot of the 18ft. ladder is exactly under this turning point. The curve generated by the top of the 18ft. ladder is irrelevant since its attitude is driven by the position of its foot and the height of the crossing point.

It seems that this is my own personal discussion thread...............
Well at least I have somebody interesting to talk to!
I have posted an illustration of the sweep of the 12ft. ladder on my web site also.
http://bernardjohnsonmusic.com/ladder_solution.htm

I'm trying to keep up with the mathematics -Bernard!

I'm not sure what you mean by a 'complex curve' - if the base of the ladder is fixed, which it is, then the tip of the ladder can only move through a regular arc.

Take a look at the diagram I have added Trevor. The foot of the ladder can only travel in a straight line. As it moves, the body of the ladder has to slide over the point. It does not rotate on the point, which it would have to do if the top (and then consequently the bottom) swung through a regular arc.
It is of further interest (to me at least) that the top of the ladder moves to the right but then moves to the left, while the foot of the ladder only moves in one direction (to the right). A reciprocating motion is generated from straight line motion.

Thanks Bernard, of course you're correct. I'd was visualising a single ladder and had forgotten about the other which of course will have to travel 'across' the snicket to accomodate the 'cross-over' point.
very impressed with the CAD demo.

I've just emailed 3 PDFs to Chris with the mathematical solution to the ladder problem - to 14 decimal places - but I'm sure the Town Planner would approximate it to the nearest inch!!
I'd like to claim credit but I can't - I passed the problem on to a bluegrass banjo picking friend of mine who relishes this sort of thing - I'm pretty impressed.

Bernard, I wanted to show your CAD solution to a friend of mine but it appears to have been deleted from your website - or have you moved it somewhere else?

Trevor,
Just looked at 3 page solution. Very impressive! Did you figure this out yourself?
If you contact me at
mushypeas4me@gmail.com
I will send you a video of the CAD solution.
Anyone else is welcome to contact me at this same address.
Bernie.

Trevor. Just returned to this thread after many months absence. Tries to access www.kbgs/ladders.pdf but all I got was an advert for viagra. I guess the ladders are no longer standing.
I encountered this problem many years ago. A girlfriend in the sixth form was irritated by a Maths teacher giving her a problem to solve each weekend so I gave her one for the teacher (the ladders). He finally solved it after about three weeks and she was miraculously freed from further weekend maths problems. The benefit to me was short-lived for she subsequently went off with a drummer.

Pity the poor drummer - much maligned and slandered - for (s)he has to sit through the failings and wailings of the "front-line" AND face the sufferers. Such attitudes towards him/her from their pretentiously musical "superiors" would fail in any hearing where the legally established tests of unlawful discrimination are applied. Notably these are where there is discrimination on the grounds of: sex; race; disability; sexual orientation or age (and who knows what yet may be added to this list?). And yet how baseless - without rhythm backing - especially from drums - are the antics (vocal or otherwise) of the posturing, vocalising, lyric-pushing "wannabes" who assail the microphone as if it were some needy charity.Drummers are the engine-room of anything musical that "moves" - and deserve any chances they may get with upwardly nubile birds.

Terry, as a bass player, I wholeheartedly agree with you. I was driving behind a van with a great bumper sticker, "I'm a drummer, that's why you are following me"