a pole leans away from the sun at an angle of 6degreesm to the vertical. when the elevation of the sun is 53degrees, the pole casts a shadow of 41feet long on level ground. how long is the pole.

To find the length of the pole, let's break down the information given:

1. The pole leans at an angle of \(6^\circ\) from the vertical.
2. The sun’s elevation is \(53^\circ\).
3. The shadow of the pole on the ground is \(41\) feet.

### Step-by-Step Solution

1. **Identify the angles in the triangle**:
   - Since the pole is tilted \(6^\circ\) from the vertical, the angle between the pole and the horizontal shadow is \(90^\circ - 6^\circ = 84^\circ\).
   - The angle of elevation of the sun is \(53^\circ\), which is the angle between the line from the top of the pole to the tip of the shadow and the horizontal ground.

2. **Set up the triangle**:
   The situation forms a triangle with:
   - One side representing the shadow on the ground (\(41\) feet),
   - An angle of \(84^\circ\) between the pole and the shadow,
   - An angle of \(53^\circ\) between the shadow and the hypotenuse (line from the top of the pole to the end of the shadow).

3. **Find the remaining angle**:
   Using the fact that the sum of angles in a triangle is \(180^\circ\):
   \[
   \text{Third angle} = 180^\circ - 84^\circ - 53^\circ = 43^\circ
   \]

4. **Use the Law of Sines**:
   Let \(L\) be the length of the pole. According to the Law of Sines:
   \[
   \frac{L}{\sin(53^\circ)} = \frac{41}{\sin(43^\circ)}
   \]
   
5. **Solve for \(L\)**:
   Rearranging the equation:
   \[
   L = \frac{41 \cdot \sin(53^\circ)}{\sin(43^\circ)}
   \]

6. **Calculate the length of the pole**:
   Using approximate values for \(\sin(53^\circ) \approx 0.7986\) and \(\sin(43^\circ) \approx 0.6820\):
   \[
   L \approx \frac{41 \cdot 0.7986}{0.6820} \approx 47.98 \text{ feet}
   \]

### Answer
The length of the pole is approximately **48 feet**.

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Would you like a more detailed breakdown of each calculation step or have further questions?

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#### Related Questions:
1. How would the length of the pole change if it leaned at a different angle?
2. How can we determine the length of the shadow if the angle of the sun changes?
3. How would the problem change if the ground was not level?
4. What if the pole was standing vertically? How would that affect the shadow?
5. How can we use trigonometric functions to verify the shadow length with different pole heights?

#### Tip:
In problems involving shadows, the Law of Sines and Law of Cosines are often helpful when dealing with non-right-angled triangles.

A pole leans away from the sun at an angle of 6 degrees to the vertical. When the elevation of the sun is 53 degrees, the pole casts a shadow of 41 feet long on level ground. How long is the pole?

Learn how to calculate the length of a pole leaning at 6 degrees to the vertical, with a sun elevation of 53 degrees casting a 41-foot shadow. This solution explores trigonometric concepts like the Law of Sines and angles in non-right triangles, suitable for students in Grades 10-12.

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