It is convenient to consider the ladder in two parts. If α is the angle the ladder makes with the ground, then the length of the ladder from the ground to the fence is
... 8/sin(α)
and the length of the ladder from the fence to the building is
... 28/cos(α)
The total ladder length (L) is the sum of these:
... L = 8/sin(α) + 28/cos(α)
The derivative of this with respect to α will be zero at the minimum length.
... dL/dα = -8cos(α)/sin(α)² +28sin(α)/cos(α)² = 0
Multiplying by the product of the denominators, we get
... 0 = -8·cos(α)³ +28·sin(α)³
Adding the cosine term and dividing by 28, we get
... (8/28)·cos(α)³ = sin(α)³
... 8/28 = tan(α)³
... α = arctan(∛(8/28)) ≈ 33.37°
The corresponding ladder length is
... L = 8/sin(33.37°) + 28/cos(33.37°)
... L ≈ 48.072 . . . . feet
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Note that the ratio 8/28 is the tangent of the angle of elevation of the fence top from the base of the building. This solution is generic in that the shortest ladder's angle with the ground is always the arctangent of the cube root of the tangent of the angle to the top of the fence from the building.
