a square fence 9 feet high enclosing a radio tower is 3 feet from the tower on all 4 sides. guy wires used to hold the tower in position are attached to the tower 30 feet above the ground and 13 feet from the fence on all 4 sides. How long are the guy wires to the nearest foot.
To find the length of the guy wires, we can use the Pythagorean theorem.
1. First, let's calculate the distance from the tower to the top corner of the fence. Since the fence is square and 9 feet high, the diagonal from the base to the top corner can be found by using the Pythagorean theorem.
a^2 + b^2 = c^2
Where a = 3 feet (distance from the tower to the side of the fence)
b = 9 feet (height of the fence)
3^2 + 9^2 = c^2
9 + 81 = c^2
90 = c^2
c ≈ 9.49 feet
2. Next, let's calculate the distance from the tower to the point where the guy wire is attached. According to the problem, this distance is 30 feet above the ground and 13 feet from the fence on all four sides. We can visualize this as the hypotenuse of a right triangle, with one leg measuring 30 feet (height above the ground) and the other leg measuring 13 feet (distance from the fence).
a^2 + b^2 = c^2
Where a = 13 feet (distance from the fence)
b = 30 feet (height above the ground)
13^2 + 30^2 = c^2
169 + 900 = c^2
1069 = c^2
c ≈ 32.69 feet
3. Since the guy wire forms a right triangle with the tower, the distance from the tower to the top corner of the fence and the distance from the tower to the point where the guy wire is attached can be considered as the two legs of the triangle.
Using the two leg lengths we calculated above, we can find the length of the guy wire by applying the Pythagorean theorem again.
a^2 + b^2 = c^2
Where a = 9.49 feet (distance from the tower to the top corner of the fence)
b = 32.69 feet (distance from the tower to the point where the guy wire is attached)
9.49^2 + 32.69^2 = c^2
90.01 + 1069.06 = c^2
1159.07 = c^2
c ≈ 34.04 feet
Therefore, the length of the guy wires to the nearest foot is approximately 34 feet.