This is a classic optimization problem involving a right triangle formed by the ladder, the fence, and the building.

We can solve it using geometric relationships and calculus.

Problem Breakdown:

• The ladder leans against the building.
• The ladder must clear an 8-foot fence located 4 feet away from the building.

Let:

• $L$ be the length of the ladder.
• $x$ be the distance from the base of the ladder to the building (horizontal distance).
• The fence is 8 feet tall and 4 feet from the building.

We need to find the length of the shortest ladder $L$ that can go over the fence and reach the building.

Step-by-Step Approach:

1. Equation for the ladder:

   • We can apply the Pythagorean theorem to express the length of the ladder.
   • The horizontal distance from the ladder’s base to the wall is $x + 4$ (since the ladder is placed beyond the fence).
   • The vertical height from the ground to the point where the ladder touches the wall is $h$.

2. Relating the ladder to the height and distance: From the similar triangles:

   • Height of the ladder over the fence: ( h = \frac{8}{4} \cdot x for_> >}