Moving a ladder around a corner

[MSD]

A ladder with length L is being moved around a corner from a hallway with a width of B to a hallway with a width of A. The ceiling has a height of K.

What is the maximum length of the ladder (maximum value of L) that can negotiate the turn?

[dimbulb]

ahhhh.. hard one.. .mainly because my differentiation and trig. skills haven't been used in YEARS!!!

lets see....(highlight to read.... )
Spoiler:
define x to be the angle between the ladder and the vertical wall. (that is, the wall that belongs to corridor B.

Let L be the maximum length of ladder than can be placed in the corner such that it has angle x with the B wall.

doing some simple trig,
L = A /cos(x) + B/sin(x) equation(1)

Now we find the angle x, for which L is minimized. (this restricts the maximum length of ladder....

dL/dx = A secx tanx -B cosecx tan x = 0
solving,
tan^3(x) = B/A

solve for x, plug this into the equation(1) to get the maximum length L.

Now, the above solution assumes the ladder is parallel to the floor. we can increase the length further by tilting the ladder so it touches the ceiling and floor.

looking from the side, it looks like a right triangle, with L and K forming the right angle.
By Pythagora's Theorem,

L" = sqrt(L^2 + K^2)

[dimbulb]

so is this correct???

[Shpoop]

i got the impression that k just meant ceiling isnt changing, but u may be right, he may have intended for u to use it

[cheerios]

if (L < K)
{ turn the laddar upright and navigate as usual; }
else
{ run crying to mommy; }

[MSD]

Yes, that looks like the right answer. Good work.

[Bloodslick]

cheerios makes me smile. She also presents the answer I'd give and use.