To solve the problem, we need to calculate the length of the guy wire, which forms the hypotenuse of a right triangle. The situation involves a 105-foot tall tower on a mountain inclined at 32° to the horizontal, and the guy wire is attached 55 feet down the slope from the base of the tower.

Here's the step-by-step approach to solve the problem:

Step 1: Label the triangle components

• The height of the tower is 105 ft (this forms one side of the right triangle, opposite the angle of the incline).
• The horizontal distance from the base of the tower to the anchor point is 55 feet, but we need to account for the slope of the mountain, which creates an inclined distance.
• The angle between the base of the tower and the horizontal is 32°.

Step 2: Apply the Law of Cosines

The guy wire forms a triangle with the tower and the slope of the mountain. We use the Law of Cosines to find the length of the wire (hypotenuse).

Let:

• $c$ be the length of the wire,
• $a = 105 \, \text{ft}$ (the height of the tower),
• $b = 55 \, \text{ft}$ (the distance down the slope),
• $\theta = 32^\circ$ (the angle between the tower and the slope).

The Law of Cosines states: $c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)$ Substituting the values: $c^2 = 105^2 + 55^2 - 2(105)(55) \cdot \cos(32^\circ)$

Step 3: Calculate the length of the guy wire

Let me calculate the value of $c$.The shortest length of the guy wire needed is approximately 65 feet.

Do you need any more details or have any questions?

Here are 5 related questions you could explore:

1. How would the length of the wire change if the slope of the mountain were steeper (e.g., a 40° incline)?
2. What if the tower height were increased to 150 feet? How would the wire length be affected?
3. How can we calculate the angle between the guy wire and the mountain slope?
4. What would be the length of the wire if it was attached 75 feet downhill instead of 55 feet?
5. How does the Law of Cosines differ from the Pythagorean Theorem, and when should each be used?

Tip: The Law of Cosines is especially useful in triangles where the sides are not perpendicular, like in this case where the mountain is inclined at an angle.