The problem you shared involves finding the shortest length of a guy wire needed to anchor the top of a tower located on an inclined mountain. The tower height is given as 213 feet, and the point where the wire is anchored is 63 feet downhill from the base of the tower, while the mountain is inclined at 36° to the horizontal.

This setup forms a right triangle with:

• The vertical height of the tower (213 feet),
• The distance along the ground from the base to the anchor point (63 feet),
• And the wire forming the hypotenuse of the right triangle.

To solve for the shortest length of the wire (the hypotenuse), we can use the Pythagorean theorem.

The Pythagorean theorem states: $c^2 = a^2 + b^2$ Where:

• $c$ is the length of the hypotenuse (the wire),
• $a$ is the height of the tower (213 ft),
• $b$ is the horizontal distance (63 ft).

Let's plug in the values and calculate the length of the wire:

$c^2 = 213^2 + 63^2$ $c^2 = 45369 + 3969 = 49338$ $c = \sqrt{49338} \approx 222.16$

Thus, the shortest length of the guy wire is approximately 222.16 feet.

Would you like more details on the steps or further explanation? Let me know if you have any questions!

Here are some related questions:

1. How does the Pythagorean theorem work in 3-dimensional problems?
2. What happens if the incline angle was steeper?
3. How would you calculate the angle of the wire if you had the lengths of the sides?
4. How does this setup change if the mountain was not inclined?
5. What are some other real-world applications of the Pythagorean theorem?

Tip: Always check if you need to use trigonometric ratios like sine, cosine, or tangent for angled problems!