A fence 8 feet tall runs parallel to a tall building at a distance of 4 ft from the building as shown in the diagram.
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We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.

[A] First, find a formula for the length of the ladder in terms of . (*Hint: split the ladder into 2 parts.*)
Type **theta** for .


To find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building, let’s analyze the geometry involved in the setup. We will split the ladder into two parts: the part extending horizontally from the wall to the fence, and the part reaching over the fence to the ground.

Let:
- \(\theta\) be the angle between the ladder and the ground.
- \(L\) be the length of the ladder we need to find.

We can split \(L\) into two segments:
1. The horizontal segment from the wall to the top of the fence, which we’ll call \(x\).
2. The hypotenuse from the top of the fence down to the ground, which we’ll call \(L_2\).

The geometry of the situation gives us two main relationships:

1. **For the segment reaching from the ground to the top of the fence (diagonal segment):**
   The ladder forms a right triangle with the ground and the height of the fence. Using trigonometry:
   \[
   \tan(\theta) = \frac{8}{4}
   \]
   where 8 feet is the height of the fence, and 4 feet is the horizontal distance from the building to the fence.



A fence 8 feet tall runs parallel to a tall building at a distance of 4 ft from the building. We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.

Solve for the shortest ladder length to reach from the ground over an 8-foot fence to a building 4 feet away. This trigonometry problem covers optimization, right triangle geometry, and uses key formulas like tan(θ) and the Pythagorean Theorem. Ideal for students in Grades 11-12.

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