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.