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This problem involves finding the length of the pole \( AC \) that leans at an angle from the vertical. Here's a breakdown of the information provided:

1. **Given:**
   - The angle of elevation of the sun is \( 53^\circ \).
   - The pole leans at an angle of \( 5^\circ \) from the vertical.
   - The shadow cast by the pole (side \( AB \)) is \( 41 \) feet.

2. **Diagram Setup:**
   - In triangle \( ABC \):
     - \( \angle A = 53^\circ \) (angle of elevation of the sun).
     - \( \angle C = 5^\circ \) (angle between the pole and the vertical).
     - Side \( AB = 41 \) feet (length of the shadow).

3. **Objective:**
   - To find the length of the pole, \( AC \).

4. **Solution Approach:**
   - We can use the **Law of Sines** in triangle \( ABC \) to solve for \( AC \).
   - First, calculate \( \angle B \), since the angles in a triangle sum up to \( 180^\circ \):
     \[
     \angle B = 180^\circ - 53^\circ - 5^\circ = 122^\circ
     \]
   - Using the Law of Sines:
     \[
     \frac{AC}{\sin(53^\circ)} = \frac{41}{\sin(122^\circ)}
     \]
   - Rearranging to solve for \( AC \):
     \[
     AC = \frac{41 \cdot \sin(53^\circ)}{\sin(122^\circ)}
     \]

5. **Calculation:**
   - Using approximate values:
     - \( \sin(53^\circ) \approx 0.7986 \)
     - \( \sin(122^\circ) \approx 0.8480 \)
   - Substitute these values:
     \[
     AC = \frac{41 \cdot 0.7986}{0.8480} \approx \frac{32.303}{0.8480} \approx 38.1
     \]

Therefore, the length of the pole \( AC \) is approximately **38 feet**.

Would you like a deeper breakdown of these steps or have any further questions?

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### Related Questions:
1. How would this problem change if the angle of the pole from the vertical were larger?
2. Can the Law of Cosines also be used to solve this triangle?
3. How would you solve for \( AB \) if only \( AC \) were given?
4. What are some practical applications of using the Law of Sines in real life?
5. How does the length of the shadow change if the sun’s elevation angle is different?

### Tip:
For solving triangles, knowing the Law of Sines and Law of Cosines can be invaluable, especially in cases where two angles and one side are known, or two sides and one angle are known.

A pole leans away from the sun at an angle of 5° to the vertical. When the elevation of the sun is 53°, the pole casts a shadow 41 ft long on level ground. How long is the pole?

Learn how to determine the length of a pole leaning away from the sun at a 5° angle to the vertical, with a shadow cast 41 ft long on level ground and the sun's elevation angle at 53°. This problem uses trigonometry and the Law of Sines, suitable for high school students in Grades 10-12.

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