A fence 8 feet tall runs parallel to a tall building at a distance of 4 feet from the building. What is the length (in feet) of the shortest ladder that will reach from the ground over the fence to the wall of the building? (Round your answer to two decimal places.)

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_> >}

Find the length of the shortest ladder that can reach over an 8-foot fence standing 4 feet away from a building. This problem applies geometric concepts and optimization techniques using calculus. Suitable for advanced high school and college-level students.

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