A fence 5 feet tall runs parallel to a tall building at a distance of 6 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.
Here are some hints for finding a solution:
Use the angle that the ladder makes with the ground to define the position of the ladder and draw a picture of the ladder leaning against the wall of the building and just touching the top of the fence.

If the ladder makes an angle 0.65 radians with the ground, touches the top of the fence and just reaches the wall, calculate the distance along the ladder from the ground to the top of the fence.

To solve this problem, we can break it down into steps based on the given parameters.

### Step 1: Understanding the Geometry

1. **Ladder Angle**: The angle the ladder makes with the ground is \( \theta = 0.65 \) radians.
2. **Height of the Fence**: The fence is 5 feet tall.
3. **Distance from the Building to the Fence**: The distance from the building to the fence is 6 feet.

### Step 2: Using Trigonometry

The ladder forms a right triangle with the ground and the wall. We can use trigonometric relationships to find the length of the ladder segment that reaches to the top of the fence.

Let \( L \) be the length of the ladder, \( h_f \) be the height of the fence (5 feet), and \( d \) be the horizontal distance from the base of the ladder to the base of the fence (which is 6 feet).

From the right triangle formed by the ladder, the height it reaches can be expressed as:
\[
h_f = L \sin(\theta)
\]
And the horizontal distance can be expressed as:
\[
d = L \cos(\theta)
\]

### Step 3: Setting up the Equations

Using the horizontal distance:
\[
d = L \cos(0.65)
\]

And we know:
\[
d = 6 \text{ feet}
\]

Setting the two equations equal gives us:
\[
6 = L \cos(0.65)
\]

From this, we can solve for \( L \):
\[
L = \frac{6}{\cos(0.65)}
\]

### Step 4: Calculating \( L \)

Now, we will calculate \( L \):
\[
\cos(0.65) \approx 0.7937 \quad (\text{using a calculator})
\]

Thus:
\[
L = \frac{6}{0.7937} \approx 7.558 \text{ feet}
\]

### Conclusion

The length of the ladder that will reach from the ground over the fence to the wall of the building, making an angle of 0.65 radians with the ground, is approximately **7.56 feet**.

### Additional Questions

1. What is the vertical height the ladder reaches on the wall?
2. How does changing the angle of the ladder affect its length?
3. What would be the length of the ladder if the fence height increased to 7 feet?
4. How would the position of the ladder change if the fence were moved closer to the building?
5. What are the implications of using different angles for the ladder's position?

### Tip

When working with problems involving angles and distances, always sketch a diagram to visualize the relationships between the components. This can greatly aid in understanding the problem and setting up equations correctly.

A fence 5 feet tall runs parallel to a tall building at a distance of 6 feet from the building. We want to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building. If the ladder makes an angle 0.65 radians with the ground, touches the top of the fence and just reaches the wall, calculate the distance along the ladder from the ground to the top of the fence.

Explore how to calculate the shortest ladder length needed to reach a building over a 5-foot fence located 6 feet away. This problem involves trigonometric functions and right triangle relationships, suitable for high school students.

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