Answer:
10.81 ft
Explanation:
You want the length of the shortest ladder that will reach over a 2 ft fence to a building 6 ft beyond.
Geometry
Where the angle the ladder makes with the ground is α the distance from the base of the ladder to the top of a fence of height h is ...
to fence = h/sin(α) = d·csc(α)
The ladder length from the fence top to the building at distance d is ...
to building = d/cos(α) = d·sec(α)
The total length of the ladder is ...
L = h·csc(α) +d·sec(α)
Minimum
The length will be minimum when its derivative with respect to α is zero.
dL/dα = -h·csc(α)cot(α) +d·sec(α)tan(α) = 0
tan(α)³ -h/d = 0 . . . . . . . . . divide by d·csc(α)cot(α)
α = arctan(∛(h/d)
For the given distance and height, the optimum angle is ...
α = arctan(∛(2/6)) ≈ 34.74°
Then the ladder length is ...
L = 6·csc(34.74°) +2·sec(37.74°) ≈ 10.81 . . . . ft
The length of the shortest ladder that will reach over the fence to the building is 10.81 feet.