A 100 foot vertical tower is to be erected on a side of a hill that makes a 6 degree angle with the horizontal. Find the length of each of the two guy wires that will be anchored 75 feet uphill and downhill from the base of the tower.
To solve this problem, we can use trigonometry. Let's start by visualizing the problem.
We have a tower that is 100 feet tall, and it is erected on the side of a hill. The hill makes a 6 degree angle with the horizontal. We want to find the length of two guy wires that will be anchored 75 feet uphill and downhill from the base of the tower.
First, let's break down the problem into simpler components. We can create a right triangle with the tower acting as the vertical side, the hill acting as the hypotenuse, and the base acting as the horizontal side.
Let's label the hypotenuse as "h" (which we will solve for later), the base as "b" (which is 75 feet), and the tower as "t" (which is 100 feet).
Now, we can use trigonometry to find the length of the hypotenuse (h). We will use the sine function because we have the opposite side (t) and the hypotenuse (h).
sin(θ) = t / h
Given that the angle θ is 6 degrees and t is 100 feet, we can rearrange the equation to solve for h:
h = t / sin(θ)
h = 100 / sin(6°)
Using a calculator, we find that h ≈ 962.46 feet.
Now that we have the length of the hypotenuse (h), we can find the length of each guy wire. Since the guy wires are anchored 75 feet uphill and downhill from the base of the tower, the length of each guy wire will be the hypotenuse of a right triangle where the base is 75 feet and the height is h.
Using the Pythagorean theorem:
(Length of guy wire)^2 = (base)^2 + (height)^2
For the uphill guy wire:
(Length of uphill guy wire)^2 = 75^2 + 962.46^2
For the downhill guy wire:
(Length of downhill guy wire)^2 = 75^2 + 962.46^2
Using a calculator, we can calculate the lengths of each guy wire:
For the uphill guy wire:
Length of uphill guy wire ≈ √(75^2 + 962.46^2)
For the downhill guy wire:
Length of downhill guy wire ≈ √(75^2 + 962.46^2)
Thus, the length of each of the two guy wires, anchored 75 feet uphill and downhill from the base of the tower, can be calculated using the above formulas.