Two hallways meet at a right angle. One hallway is w units wide and the other is v units wide. What is the length of the longest ladder which can go around this corner?

Question

The general answer is messy! But if w = v, you can show that the answer is 2w√2.

Answer 1

Let L =L₁ + L₂ where L₁ = wcscΘ and L₂ = vsecΘ where Θ is the angle one of the ends of the ladder makes with a wall. So now L = wcscΘ + vsecΘ. Take the derivative of L wrt Θ to dL/dΘ = -wcscΘctnΘ +vsecΘtanΘ. Set dL/dΘ to 0 and get w/sinΘtanΘ = vtanΘ/cosΘ. Simplifying this gives w/v = tan³Θ and tanΘ =(w/v)^1/3 use

secΘ = (1 + tan²Θ)^1/2 and cscΘ = (1 + ctn²Θ)^1/2 ctnΘ = (w/v)^-1/3 and so plugging in the expressions into the equation for L gives L =w(1 + (w/v)^-2/3)^1/2 + v(1 + (w/v)^2/3)^1/2 This gives L= 2w √2 when w=v.

Answer 2

Great job!

Not the way I solved it, but correct & very neat!

Thanks!