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 break it down into several steps:

Step 1: Visualize the Problem
Draw a diagram to help visualize the situation. On the diagram, you should have a horizontal line representing the ground, a vertical line representing the tower, and another line that connects the tower to the ground at an angle of 6 degrees.

Step 2: Identify Key Information
From the problem statement, we are given that the tower has a height of 100 feet, and the guy wires are anchored 75 feet uphill and downhill from the base of the tower. The angle between the ground and the guy wire is 6 degrees.

Step 3: Determine the Length of the Guy Wires
To find the length of each guy wire, we can use trigonometry. In this case, we can use the tangent function because we know the opposite side (100 feet) and want to find the adjacent side (length of the guy wire).

Step 4: Calculate the Lengths
Let's start with the uphill guy wire:
- Using the tangent function: tan(6 degrees) = opposite / adjacent
- Plugging in the values: tan(6 degrees) = 100 feet / length of uphill guy wire (let's call it x)
- Multiply both sides by x: x * tan(6 degrees) = 100 feet
- Solve for x by dividing both sides by tan(6 degrees): x = 100 feet / tan(6 degrees)

Now let's calculate the length of the downhill guy wire:
- Using the same tangent function: tan(6 degrees) = opposite / adjacent
- Plugging in the values: tan(6 degrees) = 100 feet / length of downhill guy wire (let's call it y)
- Multiply both sides by y: y * tan(6 degrees) = 100 feet
- Solve for y by dividing both sides by tan(6 degrees): y = 100 feet / tan(6 degrees)

Step 5: Evaluate the Results
Now that we have the equations for both the uphill and downhill guy wires, we can calculate their respective lengths using a scientific calculator or an online trigonometric calculator. Plugging in the values, we get:
- Length of uphill guy wire: x = 100 feet / tan(6 degrees)
- Length of downhill guy wire: y = 100 feet / tan(6 degrees)

After evaluating these equations, you will have the lengths of both the uphill and downhill guy wires.