https://file.aiquickdraw.com/mathbot/file/1730200661921lio5o1vn.jpg

To solve this problem, let’s analyze the given information and the diagram:

1. **Given data**:
   - The pole leans at an angle of \( 8^\circ \) from the vertical.
   - The elevation angle of the sun is \( 51^\circ \).
   - The length of the shadow cast by the pole is \( 44 \) feet.

2. **Objective**:
   - Find the length of the pole, denoted as \( AC \).

3. **Explanation**:
   In the triangle \( ABC \):
   - \( \angle BAC = 51^\circ \), which is the angle of elevation of the sun.
   - \( \angle ACB = 8^\circ \), which is the tilt angle of the pole from the vertical.
   - \( AB = 44 \) feet is the length of the shadow.

   Since the pole leans away from the sun, we can find the length \( AC \) by using the Law of Sines.

4. **Using the Law of Sines**:
   The interior angle at \( C \) of \( \triangle ABC \) can be calculated as:
   \[
   \angle ACB = 180^\circ - 51^\circ - 8^\circ = 121^\circ
   \]

   Applying the Law of Sines:
   \[
   \frac{AC}{\sin(51^\circ)} = \frac{AB}{\sin(121^\circ)}
   \]

   Rearranging to solve for \( AC \):
   \[
   AC = \frac{44 \cdot \sin(51^\circ)}{\sin(121^\circ)}
   \]

5. **Calculating**:
   Let's plug in the values (using approximate values for the sines):
   - \( \sin(51^\circ) \approx 0.7771 \)
   - \( \sin(121^\circ) \approx 0.8572 \)

   Then,
   \[
   AC = \frac{44 \times 0.7771}{0.8572} \approx \frac{34.1914}{0.8572} \approx 39.88
   \]

   Rounding to the nearest foot, the length of the pole \( AC \) is approximately **40 feet**.

The answer is:
\[
\text{The pole is } 40 \text{ feet long.}
\]

Would you like further details or have any questions?

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**Related Questions**:

1. How does the Law of Sines help in solving triangle-related problems?
2. What other methods could be used to solve for \( AC \) if different information were given?
3. How would the problem change if the angle of elevation of the sun were different?
4. How does the tilt angle of the pole affect the length of its shadow?
5. What role do interior angles of a triangle play in solving such problems?

**Tip**: Remember, the Law of Sines is particularly useful for non-right triangles where at least one angle and its opposite side are known.

A pole leans away from the sun at an angle of 8° to the vertical. When the elevation of the sun is 51°, the pole casts a shadow 44 ft long on level ground. How long is the pole?

Solve a trigonometry problem involving a leaning pole that casts a shadow. Given angles of elevation and tilt, and the shadow length, use the Law of Sines to determine the pole's length. Suitable for high school students.

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