To solve this problem, we can use the Pythagorean theorem since this forms a right triangle. Here's how we can break it down:

• The height of the tower is one leg of the right triangle, which is 80 m.
• The horizontal distance from the base of the tower to where the guy-wire meets the ground is the other leg, which is 165 m.
• The guy-wire is the hypotenuse of the triangle, which we are trying to find.

Using the Pythagorean theorem: $a^2 + b^2 = c^2$ where:

• $a = 80 \, \text{m}$ (height of the tower),
• $b = 165 \, \text{m}$ (horizontal distance from the tower base),
• $c$ is the length of the guy-wire.

Now, substitute the known values: $80^2 + 165^2 = c^2$ $6400 + 27225 = c^2$ $33625 = c^2$ To find $c$, take the square root of both sides: $c = \sqrt{33625} \approx 183.5 \, \text{m}$

So, the length of the guy-wire is approximately 183.5 meters.

Would you like further explanation or have any questions on this? Here are some related questions you might find interesting:

1. What would happen if the distance downhill was shorter, say 100 meters?
2. How would you adjust the height of the tower if the guy-wire needs to be exactly 200 meters long?
3. Can we apply the Pythagorean theorem to more complex structures involving more wires?
4. How does the angle of the guy-wire relate to the dimensions of the triangle?
5. What if the ground was not flat, but instead curved? Would the Pythagorean theorem still work?

Tip: Always check the units you're using in word problems to make sure they are consistent across the problem, and remember to square both numbers when using the Pythagorean theorem!