Answer:
16.65 ft
Explanation:
You want the length of the shortest ladder that will reach a building over an 8 ft high fence that is 4 ft from the building.
Similar triangles
As in the attached diagram, we can define the length of segment AX from the fence to the ladder base as 'x'. Then the length of the ladder to the top of the fence is found using the Pythagorean theorem to be ...
BX = √(x² +8²)
The remaining length of the ladder is the hypotenuse of a triangle similar to ∆BAX. The scale factor is DA/AX = 4/x, so the length of the remaining ladder is ...
CB = (4/x)BX = (4/x)√(x² +8²)
Ladder length
The total ladder length is the sum of its parts:
CX = CB +BX
CX = (4/x)√(x² +8²) +√(x² +8²)
CX = (1 +4/x)√(x² +8²)
Minimum length
The minimum length will be that associated with the value of x that makes the derivative of CX be zero. The second attachment shows the derivative of the total length function in terms of generic distances DA=d and BA=h. For this problem, where (d, h) = (4, 8), the derivative is ...
CX' = (1+4/x)x/√(x² +8²) -(4/x²)√(x² +8²)
Expressing this over a common denominator, we have ...
CX' = (x³ -4·8²)/(x²√(x²+8²))
This is zero when ...
x³ -4·8² = 0 ⇒ x = 4∛4 ≈ 6.3496
Total length
Using this value in the ladder length formula above, we find the length of the ladder to be ...
CX = (1 +4/6.3496)√(6.3496² +8²) ≈ 16.64775
The length of the shortest ladder is about 16.65 feet.