The problem describes a 550-foot tower on a hill with a $6^\circ$ incline, and it requires calculating the length of the uphill guy wire. The wire is anchored 70 feet uphill from the base of the tower.

We can approach this using trigonometry. Here's how:

Given:

• Height of the tower ($h$) = 550 feet.
• Distance uphill ($d$) = 70 feet.
• Hill angle with the horizontal ($\theta$) = $6^\circ$.

The total distance between the base of the tower and the guy wire's uphill anchor point is the horizontal distance (70 feet) plus the vertical projection of the height due to the hill slope.

Step 1: Horizontal component of the tower's height

To adjust for the angle of the hill, we need to project the height of the tower onto the sloped hill. This uses trigonometric ratios.

The total horizontal distance from the top of the tower to the guy wire's anchor point is given by: $\text{Horizontal distance} = 70 + h \cdot \tan(\theta)$ where $h = 550$ feet and $\theta = 6^\circ$.

Step 2: Use Pythagoras to calculate the length of the wire

The guy wire forms a right triangle with the horizontal distance and the tower's height. The length of the guy wire can be calculated using the Pythagorean theorem: $L = \sqrt{\text{Horizontal distance}^2 + h^2}$

Let me calculate that for you.The length of the uphill guy wire is approximately 564.7 feet.

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:

1. How would the calculation change if the hill's angle increased to 10 degrees?
2. What would be the effect on the guy wire length if the tower height increased to 600 feet?
3. Can you calculate the downhill guy wire's length based on this problem's data?
4. How does the horizontal projection of the tower height affect the guy wire length?
5. What role does the Pythagorean theorem play in problems involving inclined planes?

Tip: In trigonometry, remember that even small angles can significantly affect projections when working with large dimensions!