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 find the length of the guy wires, we can break down the problem into two right triangles: one for the uphill guy wire and one for the downhill guy wire.
Let's start by understanding the situation. We have a tower that is 100 feet tall and a hillside that makes a 6 degree angle with the horizontal. We need to find the length of the two guy wires that will be anchored 75 feet uphill and downhill from the base of the tower.
First, we need to calculate the height of the hillside. Since we know the angle and the length of the base of the hill (75 feet), we can use trigonometry to find the height.
The height of the hillside can be calculated using the formula:
height = base * sin(angle)
Given that the base is 75 feet and the angle is 6 degrees, we can substitute these values into the formula:
height = 75 * sin(6)
Using a calculator, we find that the height of the hillside is approximately 7.85 feet.
Now, let's calculate the length of the uphill guy wire. We can use the Pythagorean theorem, since we have a right triangle formed by the tower, the hillside, and the guy wire.
The length of the uphill guy wire can be calculated using the formula:
length = sqrt(tower height^2 + hillside height^2)
Substituting the values we have, we get:
length = sqrt(100^2 + 7.85^2)
Calculating this gives us a length of approximately 100.07 feet for the uphill guy wire.
Similarly, we can calculate the length of the downhill guy wire using the same formula:
length = sqrt(tower height^2 + hillside height^2)
Substituting the values, we get:
length = sqrt(100^2 + 7.85^2)
Calculating this gives us a length of approximately 100.07 feet for the downhill guy wire as well.
Therefore, the length of each of the two guy wires that will be anchored 75 feet uphill and downhill from the base of the tower is approximately 100.07 feet.