Tower Survey: Angle of Elevation and a Wire-Length Decision
A telecom company plans a mobile tower on level ground beside a straight road. A surveyor stands at point A on the road and measures the angle of elevation of the top of the tower as 30°. She walks 40 m straight towards the foot of the tower to point B and now measures the angle of elevation as 60°. The tower is vertical and its foot is on the same level ground. The company also needs a guy-wire from the top of the tower to point A; the supplier stocks wire in 70 m lengths and will not cut, so the company must know whether one length is enough. A trainee computes the height as 40 tan 60° = 69.3 m and is surprised that the answer exceeds the 40 m walk. The surveyor suspects a mistake in choosing which ratio to use.
1. Find the height of the tower and the distance from B to the foot of the tower.
Read the worked answer
Let the height be h metres and let F be the foot of the tower. Let BF = d. From B, the angle of elevation is 60°:
tan 60° = h/d, so d = h/√3.
From A, the angle is 30° and AF = AB + BF = 40 + d:
tan 30° = h/(40 + d), so 1/√3 = h/(40 + h/√3).
Cross-multiply:
√3h = 40 + h/√3.
Multiply by √3:
3h = 40√3 + h, so 2h = 40√3, giving h = 20√3 ≈ 34.64 m.
Then d = h/√3 = 20 m. Check: AF = 60 m, and tan 30° = 34.64/60 ≈ 0.577, as required. The tower is about 34.6 m high and B is 20 m from its foot.
2. Decide whether one 70 m wire from the top to A is sufficient, and explain the trainee's error.
Read the worked answer
The wire runs from the top T to A. In the right triangle TFA, the angle at A is 30° and the opposite side is h. Thus
sin 30° = h/TA, so TA = h/sin 30° = (20√3)/(1/2) = 40√3 ≈ 69.28 m.
This is just under 70 m, leaving about 0.72 m. So one 70 m length is sufficient in pure geometry, but the margin is small; allowance for tying, sag and cutting error should be checked before ordering.
The trainee's value 40 tan 60° = 40√3 ≈ 69.28 m is numerically the wire length, not the height. The error is using 40 m as the horizontal distance from B, whereas 40 m is the distance AB, and the horizontal distance from B to the foot is only 20 m. Also tan 60° = opposite/adjacent; with the correct adjacent side 20 m, h = 20 tan 60° = 20√3 ≈ 34.64 m. The trainee selected the wrong triangle and the wrong side.