A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 58°. A guy wire is to be attached to the top of the tower and to the ground, 50 m downhill from the base of the tower. The angle 𝛼 in the figure is determined to be 7°. Find the length of cable required for the guy wire. (Round your answer to the nearest meter.)

To solve this problem, we need to determine the length of the guy wire (cable) required, given the angle of inclination of the hill and the angle the guy wire makes with the vertical. Here are the steps:

1. **Understanding the problem setup:**
   - The angle of inclination of the hill is \( \beta = 58^\circ \).
   - The distance downhill from the base of the tower to the point where the guy wire is attached is \( d = 50 \) meters.
   - The angle between the guy wire and the hill is \( \alpha = 7^\circ \).

2. **Identifying the angles in the triangle:**
   - The angle between the guy wire and the horizontal ground is \( \theta = \beta + \alpha = 58^\circ + 7^\circ = 65^\circ \).

3. **Using trigonometry:**
   We need to find the length of the guy wire \( L \), which is the hypotenuse of the right triangle formed by the hill and the guy wire.

   - In this right triangle, the side adjacent to \( \theta \) (along the hill) is \( d = 50 \) meters.
   - We can use the cosine function to find \( L \):
   \[
   \cos(\theta) = \frac{d}{L}
   \]
   \[
   \cos(65^\circ) = \frac{50}{L}
   \]

4. **Calculating \( L \):**
   - Find \( \cos(65^\circ) \):
     \[
     \cos(65^\circ) \approx 0.4226
     \]
   - Solve for \( L \):
     \[
     L = \frac{50}{\cos(65^\circ)} = \frac{50}{0.4226} \approx 118.3 \text{ meters}
     \]

So, the length of the guy wire required is approximately 118 meters (rounded to the nearest meter).

Would you like further details or have any questions?

Here are some related questions for further practice:

1. How would the length of the guy wire change if the angle of inclination of the hill was 45°?
2. What if the distance downhill from the base of the tower was 60 meters instead of 50 meters?
3. How does the angle \( \alpha \) affect the length of the guy wire?
4. What if the guy wire was attached to a point 70 meters downhill from the base of the tower?
5. How would you find the height of the tower given the length of the guy wire and the angles?
6. What if the angle of inclination of the hill was 30° instead of 58°?
7. How does changing the height of the tower affect the length of the guy wire?
8. What is the effect of using a different trigonometric function (sine or tangent) to solve a similar problem?

**Tip:** Always double-check angle measurements and units when solving trigonometry problems to ensure accuracy in your calculations.

Learn how to calculate the length of a guy wire on a steep hill using trigonometric principles. This problem involves finding the length of the guy wire based on angles of inclination and depression. Practice problems included.

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