1. **State the problem:** We need to find the length of the guy wire running from the top of the cell tower to the metal stake in the ground, given a 10-foot pole supporting the wire, and distances 6 ft from stake to pole and 29 ft from pole to tower. 2. **Visualize the setup:** The guy wire forms the hypotenuse of a right triangle with the vertical side being the height difference and the horizontal side being the total horizontal distance from the stake to the tower. 3. **Calculate the total horizontal distance:** The horizontal distance from the stake to the tower is the sum of the distances from stake to pole and pole to tower: $$6 + 29 = 35 \text{ ft}$$ 4. **Calculate the vertical height of the tower:** The pole is 10 ft tall, so the vertical height from the top of the pole to the top of the tower is unknown. However, since the guy wire runs from the top of the tower to the stake, and the pole supports the wire at 10 ft, the vertical height of the tower above the ground is the height of the pole plus the vertical segment from the pole top to the tower top. Since the pole supports the wire, the vertical height difference between the tower top and the stake is the height of the tower (unknown) and the stake is at ground level (0 ft). But the problem does not give the tower height directly, so we consider the right triangle formed by the guy wire, the horizontal distance 35 ft, and the vertical height of the tower. 5. **Use the Pythagorean theorem:** Let $L$ be the length of the guy wire (hypotenuse), and $h$ be the height of the tower. The horizontal distance is 35 ft. $$L^2 = h^2 + 35^2$$ 6. **Find the height $h$ of the tower:** The pole is 10 ft tall and is placed 6 ft from the stake. The distance from the pole to the tower is 29 ft. The pole supports the wire, so the wire runs from the tower top to the stake, passing over the pole top. We can consider two right triangles: - Triangle 1: from stake to pole (6 ft horizontal) and pole height 10 ft. - Triangle 2: from pole to tower (29 ft horizontal) and height difference $h - 10$ ft. The guy wire is the sum of these two segments, but since the wire is straight, the entire length is the hypotenuse of the big triangle with base 35 ft and height $h$. 7. **Calculate the height $h$ using similar triangles or Pythagorean theorem:** The wire passes over the pole top, so the height at 6 ft from the stake is 10 ft. The slope of the wire is: $$m = \frac{h - 0}{35} = \frac{h}{35}$$ At 6 ft from the stake, the height is 10 ft, so: $$10 = m \times 6 = \frac{h}{35} \times 6$$ Solve for $h$: $$10 = \frac{6h}{35} \Rightarrow h = \frac{10 \times 35}{6} = \frac{350}{6} = 58.333...$$ 8. **Calculate the length of the guy wire:** $$L = \sqrt{h^2 + 35^2} = \sqrt{58.333^2 + 35^2}$$ Calculate inside the square root: $$58.333^2 = 3402.78, \quad 35^2 = 1225$$ $$L = \sqrt{3402.78 + 1225} = \sqrt{4627.78} \approx 68.0$$ 9. **Final answer:** The length of the guy wire is approximately 68 ft.