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 use trigonometry.
First, let's draw a diagram to visualize the situation:
```
A
|\
| \
| \
| \
| \
| \
| \
|-------\
B C D
```
In the diagram above:
- Point A represents the top of the tower
- Point B represents the base of the tower
- Points C and D represent the anchoring points of the guy wires, with C uphill and D downhill from the base of the tower
- The distance from point B to point C is 75 feet
- The distance from point B to point D is 75 feet
To find the length of the guy wires, we need to find the lengths of triangles ABC and ABD.
Now we can apply trigonometry to find the lengths of the sides of the triangles. We will use the sine function, as we have the opposite side and want to find the hypotenuse. The sine function relates the opposite side (the height of the hill) to the hypotenuse (the length of the guy wire).
For triangle ABC:
Using the sine function: sin(6°) = opposite/hypotenuse
sin(6°) = BC / 75
Cross-multiplying: BC = 75 * sin(6°)
Similarly, for triangle ABD:
Using the sine function: sin(6°) = opposite/hypotenuse
sin(6°) = BD / 75
Cross-multiplying: BD = 75 * sin(6°)
Therefore, the length of each of the two guy wires (BC and BD) is 75 * sin(6°), which you can calculate using a calculator.
To solve this problem, we can break it down into two right triangles.
First, let's consider the right triangle formed by the tower, the uphill guy wire, and the horizontal ground. The side opposite the 6 degree angle is the length of the uphill guy wire, and the side adjacent to the 6 degree angle is 75 feet. We want to find the hypotenuse (length of the uphill guy wire).
Using trigonometry, we can use the formula for the sine of an angle:
sin(angle) = opposite/hypotenuse
sin(6 degrees) = 75/hypotenuse
Solving for hypotenuse:
hypotenuse = 75/sin(6 degrees)
hypotenuse ≈ 886.70 feet
Similarly, we can do the same calculations for the downhill guy wire. In this case, the side adjacent to the 6 degree angle is still 75 feet, but the side opposite the 6 degree angle is now the length of the downhill guy wire.
Using the same formula, we have:
sin(6 degrees) = 75/hypotenuse
Solving for hypotenuse:
hypotenuse = 75/sin(6 degrees)
hypotenuse ≈ 886.70 feet
Therefore, the length of each of the two guy wires (uphill and downhill) is approximately 886.70 feet.