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?

1 answer

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