A fence 2 feet tall runs parallel to a tall building at a distance of 6 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

Let the ladder length be d
Let the base of the ladder be x feet from the fence, and hit the wall at height h.

from similar triangles,

2/x = h/(6+x)
h = 2(x+6)/x

d^2 = (6+x)^2 + h^2
= (x+6)^2 + 4(x+6)^2/x^2
= (x+6)^2 * (1 + 4/x^2)

d^2 (and therefore d) has a minimum at x = 2.9

d^2 = 116.88
d = 10.8 ft