A fence 5 feet tall runs parallel to a tall building at a distance of 3 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

Answer 1

draw a diagram. If the foot of the ladder is x feet from the fence, and the ladder reaches y feet up the wall, then we have

x/5 = (x+3)/y
y = 5 + 15/x

if the length of the ladder is z,

z^2 = (x+3)^2 + y^2
= (x+3)^2 + (5 + 15/x)^2
z = √((x+3)^2 + (5 + 15/x)^2)

dz/dx = (x+3)(x^3-75) / <a bunch of nonzero junk>

so, dz/dx=0 at x = -3 or ∛75

z^2 = (3+∛75)^2 + (5+15/∛75)^2
z = 11.19 feet