Answer:
43.24 ft
Step-by-step explanation:
You want the length L of the shortest ladder that will reach from the ground across the top of an 8 ft fence to a building 24 ft beyond the fence.
Model
The attached diagram shows a model of the geometry. We have defined segment DE from the fence to the ladder base as 'x'. The hypotenuse CE of ∆CDE is given by the Pythagorean theorem as ...
CE = √(CD² +DE²) = √(64 +x²)
With respect to ∆CDE, ∆GBC is similar, with scale factor BC/DE = 24/x. Then the full length L is ...
L = CE +GC
L = CE + (24/x)CE = √(64 +x²)·(1 +24/x)
Minimum length
The minimum value of L will be the value where its derivative with respect to x is zero.
L' = x/√(64 +x²)(1 +24/x) -24/x²√(64 +x²)
0 = (x²(x+24) -24(x² +64))/(x²√(64+x²))
0 = x³ -24·64
x = ∛1536 ≈ 11.538
For this value of x, the ladder length is ...
L = √(64 +11.538²)·(1 +24/11.538) ≈ 43.2448
The minimum length ladder is about 43.24 feet long.