To solve this problem, we’ll use trigonometry, specifically the sine and cosine functions, to find the length of the pole.

Problem Breakdown:

1. A pole is leaning at a 5° angle from the vertical.
2. The sun has an elevation of 53°, creating a 53° angle with the ground at point A.
3. The pole casts a shadow of 41 feet (distance $AB = 41 \, \text{ft}$).

Diagram Analysis:

From the right triangle in the diagram:

• Let $C$ be the top of the pole.
• $AB$ represents the shadow on the ground.
• $BC$ is the vertical component of the pole.
• We need to find the length of $AC$, the length of the pole.

Solution:

Since we are given angles and the adjacent side ($AB = 41$ feet), we can find $AC$ using trigonometry.

1. Set up the relationship using cosine (based on the angle at point $A$): $\cos(53^\circ) = \frac{AB}{AC}$

2. Rearrange to solve for $AC$: $AC = \frac{AB}{\cos(53^\circ)}$

3. Substitute values and calculate:

   • $AB = 41$ feet
   • $\cos(53^\circ) \approx 0.6018$ $AC = \frac{41}{0.6018} \approx 68.13 \text{ feet}$

4. Round to the nearest foot: $AC \approx 68 \text{ feet}$

Answer:

The pole is approximately 68 feet long.

Would you like a detailed explanation of each step, or do you have further questions?

Related Questions:

1. How would the length of the pole change if the shadow was longer or shorter?
2. How does changing the angle of elevation of the sun affect the shadow length?
3. What trigonometric function could be used if the problem provided the height instead of the shadow length?
4. Can you solve for the angle of elevation if the pole's length and shadow length are known?
5. How would you approach this problem if the pole were at a different angle from the vertical?

Tip:

In trigonometry problems involving shadows and angles, carefully identify which sides correspond to which trigonometric functions based on the given angles.