Let's break down the problem step by step:

Step 1: Initial Setup

We are given a ladder initially set at an angle of 45° with the ground.

Let:

• $L$ be the length of the ladder.
• The initial position of the top of the ladder is at a height $h_1$ from the ground, and the foot of the ladder is $x_1$ meters away from the wall.

We can use trigonometry in this initial configuration:

1. Since the angle with the ground is 45°, we know that in a right triangle formed by the ladder, the wall, and the ground:
   • The height from the ground is $h_1 = L \sin(45^\circ) = \frac{L}{\sqrt{2}}$.
   • The horizontal distance from the wall is $x_1 = L \cos(45^\circ) = \frac{L}{\sqrt{2}}$.

Step 2: After the Ladder is Pulled Away

The foot of the ladder is pulled away from the wall by 4 meters, and the top slides down by 3 meters. The new position of the ladder forms an angle of 30° with the ground.

Let:

• The final height of the top of the ladder from the ground be $h_2$.
• The new distance from the wall is $x_2 = x_1 + 4$ meters.
• The length of the ladder remains $L$, and we can again apply trigonometry:
  • $h_2 = L \sin(30^\circ) = \frac{L}{2}$.
  • $x_2 = L \cos(30^\circ) = \frac{\sqrt{3}}{2} L$.

Step 3: Relating the Initial and Final Positions

The total horizontal distance between the initial and final positions is 4 meters, so: $x_2 - x_1 = 4$ This gives us the equation: $\frac{\sqrt{3}}{2} L - \frac{1}{\sqrt{2}} L = 4$

We can now solve for $L$, the length of the ladder.

Step 4: Solving for $L$

Simplifying the equation: $\left( \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \right) L = 4$

First, we find a common denominator: $\frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}, \quad \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

So the equation becomes: $\left( \frac{\sqrt{3} - \sqrt{2}}{2} \right) L = 4$

Now, solving for $L$: $L = \frac{4 \times 2}{\sqrt{3} - \sqrt{2}} = \frac{8}{\sqrt{3} - \sqrt{2}}$

We simplify the expression for $L$, using approximate values for $\sqrt{3}$ and $\sqrt{2}$: $L \approx \frac{8}{1.732 - 1.414} = \frac{8}{0.318} \approx 25.16 \text{ meters}$

Step 5: Final Height of the Top of the Ladder

Now that we have $L$, we can calculate $h_2$, the final height of the top of the ladder: $h_2 = \frac{L}{2} = \frac{25.16}{2} \approx 12.58 \text{ meters}$

Final Answer:

• The final height of the top of the ladder from the ground is approximately $12.58$ meters.
• The length of the ladder is approximately $25.16$ meters.

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Would you like further clarification on any part of the solution or have any other questions? Here are some related questions you might find interesting:

1. How would the final height change if the angle was different?
2. What if the ladder had initially been placed at a steeper angle than 45°?
3. How could you verify the ladder's length using Pythagoras' theorem in both positions?
4. What would happen if the ladder's foot was pulled away by a different distance?
5. How does the concept of trigonometric identities help solve real-life problems like this?

Tip: When dealing with angles and distances in real-world applications like this, it's always helpful to break the problem into smaller, solvable triangles and use trigonometric functions to find missing values.