Hello,
I've modified your sketch: I added a few names.
By similar triangles you get the proportion:
27/8.....y
----- = ---
... x.......x+8
and you'll get: y = 27/8 + 27/x
The length of the ladder can be calculated by (use Pythagoran theorem):
l² = (x+8)² + (27/8 + 27/x)²
l²(x) = x² + 16x + 64 + 729/64 + 729/(4x) + 729/(x²)
If you want to know the shortest ladder possible you have to calculate a minimum. Therefore you calculate the derivative of l²(x) which has to be equal to zero: (If l has a minimum, that means l'(x) = 0 then l² has a maximum, that means (l²)'(x) = 0)
(l²)'(x) = 2x + 16 - 729/(4x²) - 729/(x³). Now (l²)'(x) = 0:
2x + 16 - 729/(4x²) - 729/(x³) = 0. Multiply by x³
2x^4 + 16x³ - 729/4 x - 1458 = 0
2x³(x + 8) - 729/4(x+8) = 0
(2x³ - 729/8)(x + 8) = 0
x = -8 this result is not very plausible with your problem
x = ³√(729/8) = 9/2
Plug in this value into the equation for l²:
l² = (9/2 + 8)² + (27/8 + 27/(9/2))²
l² = 156.25 + 5625/64 = 15625/64
Therefore the ladder has the length: l = √(15625/64) = 125/8m ≈ 15.63 m
EB