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We are given the following information:

- The pole leans at \(15^\circ\) from the vertical towards the sun.
- The shadow it casts is 10 meters long.
- The angle of elevation of the sun is \(35^\circ\).

We need to determine the length of the pole.

To solve this, we can model the situation using trigonometry. Here's the step-by-step approach:

1. **Diagram setup**: 
   - Let the length of the pole be \( L \).
   - Since the pole is leaning at \(15^\circ\) from the vertical, the effective angle between the ground and the pole’s top is \(90^\circ - 15^\circ = 75^\circ\).
   - The angle between the shadow and the line connecting the top of the pole to the end of the shadow is \(35^\circ\), as it corresponds to the angle of elevation of the sun.
   - The shadow forms the base of a right-angled triangle, with the pole’s top as the hypotenuse.

2. **Use of trigonometry**:
   - The angle between the pole and the ground is \(75^\circ\), while the angle of elevation is \(35^\circ\).
   - Using the law of sines, we can write:
     \[
     \frac{L}{\sin(35^\circ)} = \frac{10 \text{ m}}{\sin(75^\circ)}
     \]
   - Solving for \(L\):
     \[
     L = \frac{10 \cdot \sin(35^\circ)}{\sin(75^\circ)}
     \]

Let's compute that.The length of the pole is approximately \( 5.94 \) meters.

Would you like a further breakdown or explanation of the steps? Here are five related questions to deepen your understanding:

1. How does the angle of elevation influence the length of the shadow?
2. What role does the leaning angle of the pole play in determining its length?
3. How would the result change if the pole were perfectly vertical?
4. What is the significance of using the law of sines in this problem?
5. Can the shadow length be longer than the pole’s length in any situation?

**Tip:** Always ensure angles are appropriately adjusted to the reference axis (vertical or horizontal) when using trigonometric principles in real-world applications.

A pole leans 15° from the vertical towards the sun. If it casts a shadow 10 m long when the angle of elevation of the sun is 35°, determine the length of the pole.

Solve a trigonometry problem where a pole leans 15° from vertical, casting a 10 m shadow with a 35° angle of elevation of the sun. Learn how to apply the Law of Sines to find the length of the pole. Suitable for Grades 10-12.

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