A 115-ft tower is located on the side of a mountain that is inclined 32° to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 55 ft downhill from the base of the tower. Find the shortest length of wire needed. (Round your answer to the nearest foot.)


To solve this problem, we need to find the shortest length of the guy wire, which forms a triangle. We can model the situation using trigonometry.

### Step-by-Step Solution:
1. **Draw a diagram**: The tower is vertical, and the mountain slope is inclined at an angle of \(32^\circ\) to the horizontal. The guy wire connects the top of the tower to a point 55 ft downhill from the base, forming a right triangle.

2. **Identify the known values**:
   - The height of the tower: \(115\) ft.
   - The distance downhill (along the slope): \(55\) ft.
   - The angle of the mountain’s incline: \(32^\circ\).

3. **Use the Law of Cosines**: Since the wire does not form a simple right triangle (because the angle between the tower and the slope is not \(90^\circ\)), we'll use the law of cosines to find the length of the wire.

The formula for the Law of Cosines is:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
Where:
- \(a = 115\) ft (the height of the tower),
- \(b = 55\) ft (the distance downhill),
- \(C = 90^\circ - 32^\circ = 58^\circ\) (the angle between the tower and the slope).

Now, let's calculate the length of the wire.

\[
c^2 = 115^2 + 55^2 - 2 \cdot 115 \cdot 55 \cdot \cos(58^\circ)
\]

We'll compute this to find the shortest length of the wire.The shortest length of the guy wire needed is approximately **98 feet**.

Would you like more details or have any further questions?

Here are some related questions you might find helpful:
1. How does the Law of Cosines work for non-right triangles?
2. What is the significance of the angle between the tower and the mountain in this problem?
3. How do we convert between degrees and radians?
4. How would the length of the wire change if the mountain's incline was steeper?
5. How do we calculate the height of the tower if the length of the guy wire is known?

**Tip:** Always ensure angles are in the correct units (degrees or radians) when using trigonometric functions in calculations.

A 115-ft tower is located on the side of a mountain that is inclined 32° to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 55 ft downhill from the base of the tower. Find the shortest length of wire needed. (Round your answer to the nearest foot.)

Discover how to find the shortest length of a guy wire needed for a 115-ft tower located on a 32° inclined mountain. This problem involves trigonometry and the Law of Cosines, providing a detailed step-by-step solution. Suitable for students in Grades 10-12.

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