To solve this problem, we can use the concept of similar triangles formed by the ladder, the fence, and the building, as well as the Pythagorean theorem.

Diagram and Setup

Imagine a right triangle where:

• The base is the ground between the fence and the building (2 feet).
• The height is from the ground to the top of the fence (8 feet).
• The hypotenuse is the part of the ladder leaning over this base and height.

Since the ladder has to extend to the wall of the building, the total ladder forms another right triangle with:

• The base as the distance from the bottom of the fence to the building (2 feet).
• The height as the unknown we need to find.
• The hypotenuse as the total length of the ladder.

Calculations

Step 1: Find the hypotenuse of the smaller triangle using the Pythagorean theorem.

The smaller triangle has legs of 2 feet and 8 feet: $c = \sqrt{2^2 + 8^2} = \sqrt{4 + 64} = \sqrt{68}$

Step 2: Solve for the total ladder length. Since the ladder is a straight line and must extend beyond the height of the fence, we consider that it continues past the fence in a similar triangle arrangement.

Thus, the total ladder length must cover the 8 feet of fence and still extend past the fence by $\sqrt{68}$ feet to reach the building: $\text{Ladder length} = 8 + \sqrt{68}$

Calculation

$\text{Ladder length} = 8 + \sqrt{68}$ $\sqrt{68} \approx 8.2 \text{ feet}$ $\text{Ladder length} \approx 8 + 8.2 = 16.2 \text{ feet}$

Hence, the shortest ladder length needed is approximately 16.2 feet.

Would you like further details or have any questions about this solution?

Here are some related questions:

1. How does the ladder’s angle with the ground change if the fence height is increased?
2. What if the distance between the fence and the building increases? How does it affect the ladder length?
3. How would you solve this problem if the fence were shorter than 8 feet?
4. Can you describe the use of trigonometry in solving this problem instead of the Pythagorean theorem?
5. What safety considerations should be taken into account when placing a ladder against a building in this manner?

Tip: Always ensure to check the assumptions made in the calculations, such as the ladder being straight and reaching exactly to the top of the fence.