Final answer:
To find the shortest length of guy wire, we use trigonometry to calculate the horizontal distance and height of the tower above the base, then apply the Pythagorean theorem.
Step-by-step explanation:
The question involves finding the shortest length of guy wire needed to attach to the top of a located on a 35° inclined mountain and anchored 40 ft downhill from the base of the tower. To solve this problem, we need to apply the principles of trigonometry. The tower, mountain slope, and guy wire form a right-angled triangle, with the wire as the hypotenuse.
Firstly, we find the horizontal distance from the base of the tower to the guy wire's anchor point. Since the base of the triangle is 40 ft along the slope and the slope makes a 35° angle with the horizontal, we can use the cosine function:
Horizontal Distance = 40 ft × cos(35°)
Then, we calculate the height of the tower above this horizontal distance. Since the tower is perpendicular to the mountain slope, the angle between the tower and the slope is 90° - 35° = 55°. We can now use the sine function:
Height Above Base = 256 ft × sin(55°)
Finally, we find the length of the guy wire using the Pythagorean theorem:
Length of Guy Wire = √(Horizontal Distance² + Height Above Base²)