A building is 2ft from a 12-ft fence that surrounds the property. A worker wants to wash a window in the building 16ft from the ground. He plans to place a ladder over the fence so it rests against the building.​ (See the​ figure.) He decides he should place the ladder 11ft from the fence for stability. To the nearest tenth of a​ foot, how long a ladder will he​ need?

We need to find the length of the ladder required for the worker to reach the window. To do this, we'll break the problem down using the geometry of the situation.

### Key Points:
- The building is 2 feet away from the 12-foot fence.
- The ladder is placed 11 feet away from the fence.
- The worker needs to reach a window that is 16 feet high.

### Step-by-step Process:

1. **Distance from the ladder's base to the building**:
   The ladder is placed 11 feet away from the fence, and the building is 2 feet from the fence. So, the total horizontal distance between the base of the ladder and the building is:
   \[
   11 \, \text{ft} + 2 \, \text{ft} = 13 \, \text{ft}
   \]

2. **Ladder forms a right triangle**:
   The ladder forms a right triangle with the building where:
   - The vertical distance (height of the window) is 16 feet.
   - The horizontal distance (from the base of the ladder to the building) is 13 feet.
   - The length of the ladder (hypotenuse) is what we need to find.

   Using the Pythagorean theorem:
   \[
   a^2 + b^2 = c^2
   \]
   where:
   - \( a = 13 \) feet (horizontal distance),
   - \( b = 16 \) feet (vertical height),
   - \( c \) is the length of the ladder.

3. **Substitute the values and solve**:
   \[
   13^2 + 16^2 = c^2
   \]
   \[
   169 + 256 = c^2
   \]
   \[
   425 = c^2
   \]
   Taking the square root of both sides:
   \[
   c = \sqrt{425} \approx 20.6 \, \text{feet}
   \]

### Final Answer:
The worker will need a ladder that is approximately **20.6 feet** long.

Would you like further details or clarifications?

Here are 5 related questions you might find helpful:
1. How would the ladder length change if the building were farther from the fence?
2. What happens to the ladder length if the window height is increased?
3. How can we verify the stability of the ladder placement?
4. What if the fence height was not given? How would the calculation change?
5. How does the angle of elevation affect the ladder length?

**Tip**: When using the Pythagorean theorem, ensure all units are consistent and verify which sides correspond to the right-angle triangle's legs and hypotenuse.

A building is 2ft from a 12-ft fence that surrounds the property. A worker wants to wash a window in the building 16ft from the ground. He plans to place a ladder over the fence so it rests against the building. He decides he should place the ladder 11ft from the fence for stability. To the nearest tenth of a foot, how long a ladder will he need?

Learn how to calculate the length of a ladder needed to reach a window 16 feet high in a building. Using the Pythagorean theorem, we find that the ladder must be approximately 20.6 feet long, considering the ladder is placed 11 feet from a 12-ft fence with the building 2 feet away. This problem is suitable for students in Grades 8-10.

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