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 can model it using trigonometry. We need to find the shortest length of the guy wire, which will form the hypotenuse of a right triangle.

Here is a breakdown of the scenario:
- The tower is vertical, with a height of 115 feet.
- The mountain has an incline of 32° to the horizontal.
- The distance from the base of the tower to the point downhill where the wire is anchored is 55 feet.

To calculate the shortest length of the wire, follow these steps:

### Step 1: Set up the coordinate system
Let the base of the tower be at point \( A \), the top of the tower be at point \( B \), and the point downhill where the wire is anchored be point \( C \).

### Step 2: Determine the coordinates of point \( C \)
Since point \( C \) is 55 feet downhill and the slope of the mountain is inclined at 32°, we can calculate the horizontal and vertical components of this distance.
- Horizontal component = \( 55 \times \cos(32^\circ) \)
- Vertical component = \( 55 \times \sin(32^\circ) \)

So the height of point \( C \) below the base of the tower will be equal to the vertical component of this 55-foot distance.

### Step 3: Determine the total vertical distance
Since the tower is 115 feet tall, the total vertical distance from point \( C \) to the top of the tower will be:
\[
\text{Vertical distance} = 115 + (55 \times \sin(32^\circ))
\]

### Step 4: Calculate the shortest length of the wire
Using the Pythagorean theorem:
\[
\text{Length of the wire} = \sqrt{(\text{horizontal component})^2 + (\text{total vertical distance})^2}
\]
Now we can calculate these values.

I'll compute this next.The shortest length of the guy wire needed is 152 feet.

Would you like further details on any of these steps, or have any questions?

Here are five related questions to expand on this:
1. How does the angle of inclination affect the length of the guy wire?
2. What if the angle of the mountain was steeper, say 45°? How would that change the result?
3. How would the calculation change if the guy wire were anchored uphill instead of downhill?
4. Could the problem be solved using trigonometric identities instead of the Pythagorean theorem?
5. How does the slope distance affect the horizontal and vertical components?

**Tip:** When dealing with inclined surfaces, always break down distances into horizontal and vertical components to simplify the calculations.

Learn how to calculate the shortest length of a guy wire needed for a 115-ft tower on a 32° inclined mountain. This problem involves key trigonometric concepts like the Pythagorean theorem and inclined plane analysis. Suitable for students in Grades 10-12.

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