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A hill makes an angle of 20° with the horizontal.  On the hill stood an old, wobbly, wooden, 50-foot tall watch tower that was starting to lean.  A guy wire was attached to the top of the watch tower and secured to a spot 80 feet uphill.  The guy wire was tightened until the watch tower was standing perfectly vertical again.  Additional braces were added to make sure the tower wobbled no more!  How long was the guy wire?

To solve this problem, let's analyze the given scenario using trigonometry. Here's how we can proceed:

### Given:
- The hill makes an angle of \( 20^\circ \) with the horizontal.
- The watch tower is \( 50 \) feet tall and is now vertical.
- The guy wire is attached to the top of the tower and is secured \( 80 \) feet uphill.
- We need to find the length of the guy wire.

### Solution:
1. **Visualize the Triangle:**
   - The guy wire, the height of the tower, and the horizontal distance uphill form a right triangle.
   - Since the hill is inclined at \( 20^\circ \), we need to adjust the position of the tower so it stands vertical. When the guy wire is attached, it forms a triangle where:
     - The height of the triangle is the height of the tower (\( 50 \) feet).
     - The base of the triangle will be the horizontal component of the distance uphill.
     - The hypotenuse of the triangle is the guy wire (which we need to find).

2. **Decomposing the Lengths:**
   - The \( 80 \) feet uphill distance is not horizontal but along the slope of the hill. To find the horizontal component, we need to use:
     \[
     x = 80 \cos(20^\circ)
     \]
     where \( x \) is the horizontal distance from the base of the tower to where the guy wire is secured.

3. **Use Pythagoras' Theorem:**
   - The guy wire, \( L \), is the hypotenuse of the right triangle:
     \[
     L^2 = 50^2 + x^2
     \]

4. **Calculation:**
   First, calculate the horizontal distance \( x \):
   \[
   x = 80 \cos(20^\circ) \approx 75.18 \text{ feet}
   \]

   Now, use Pythagoras' theorem to find \( L \):
   \[
   L = \sqrt{50^2 + 75.18^2}
   \]
   \[
   L = \sqrt{2500 + 5651.68}
   \]
   \[
   L = \sqrt{8151.68} \approx 90.31 \text{ feet}
   \]

### Conclusion:
The length of the guy wire is approximately \( 90.31 \) feet.

Would you like more details or explanations on any part of this solution?

### Related Questions:
1. How would the solution change if the angle of the hill was \( 30^\circ \) instead of \( 20^\circ \)?
2. Can you determine the tension in the guy wire if the tower exerts a force of 500 N vertically down?
3. How would you calculate the force components acting along the hill's incline due to the guy wire?
4. What if the guy wire was attached to a point 60 feet uphill instead of 80 feet?
5. Could you determine the angle at which the guy wire makes with the hill's surface?

### Tip:
When dealing with inclined planes, always decompose vectors and distances into components parallel and perpendicular to the incline. This helps in simplifying the calculation and ensures accuracy.

A hill makes an angle of 20° with the horizontal. On the hill stood an old, wobbly, wooden, 50-foot tall watch tower that was starting to lean. A guy wire was attached to the top of the watch tower and secured to a spot 80 feet uphill. The guy wire was tightened until the watch tower was standing perfectly vertical again. Additional braces were added to make sure the tower wobbled no more! How long was the guy wire?

Solve how to find the length of a guy wire attached to a 50-foot tall watch tower on a hill with a 20° incline. The guy wire is secured 80 feet uphill. Using trigonometry and the Pythagorean Theorem, calculate the wire's length step by step. Suitable for Grades 9-12.

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