A fence 12ft high runs parallel to a tall building at a distance of 2ft 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

If the ladder of length z touches the ground x feet from the fence, and touches the building at height y, then

z^2 = (x+2)^2 + y^2

Using similar triangles we see that
x/12 = (x+2)/y
y = 12(x+2)/x

z^2 = (x+2)^2 + (12(x+2)/x)^2

So, find dz/dx and set it zero, to see that the minimum z occurs at x = ∛288. So, the shortest ladder is 17.85 feet.