A fence 6 feet tall runs parallel to a tall building at a distance of 20 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. Let
z = length of ladder
x = distance from fence to foot of ladder
θ = angle ladder makes with the ground
Then we have
tanθ = 6/x
cosθ = (x+20)/z
So,
z = (x+20)secθ = (x+20)√(1+(6/x)^2) = (x+20)/x * √(x^2+36)
dz/dx = (x^3-720)/(x^2 √(x^2+36))
dz/dx=0 when x=2∛90 ≈ 8.96
z=34.85 ft