Tower Height and Safety Wire
A surveyor is checking a vertical mobile tower standing on level ground. From point A, the angle of elevation of the top of the tower is 30°. From point B, which lies 20 m closer to the tower on the same straight line, the angle of elevation is 45°. The surveyor uses these readings to estimate the height and to plan a straight safety wire from the top of the tower to point A. A supplier offers a 55 m wire. Installation requires an extra 1.5 m of wire in total for clamps and tensioning, so the supplied length must exceed the straight-line distance by at least 1.5 m. The surveyor claims the tower is less than 25 m tall. Use √3 ≈ 1.732. The tower is vertical, the ground is level, and the readings are taken at ground level.
1. Find the height of the tower and evaluate the surveyor's claim that it is less than 25 m tall.
Read the worked answer
Let F be the foot of the tower and h = TF its height. From B, tan 45° = h/BF, so BF = h. From A, the distance to the foot is AF = BF + 20 = h + 20. From A, tan 30° = h/(h + 20). Since tan 30° = 1/√3, we get h/(h + 20) = 1/√3, so √3 h = h + 20. Hence h(√3 - 1) = 20, and h = 20/(√3 - 1). Rationalising: h = 20(√3 + 1)/((√3 - 1)(√3 + 1)) = 20(√3 + 1)/2 = 10(√3 + 1). Using √3 ≈ 1.732 gives h ≈ 10 × 2.732 = 27.32 m. The tower is about 27.3 m tall, so the surveyor's claim that it is less than 25 m is false.
2. Find the straight-line length of the safety wire from the top of the tower to point A. Is the supplier's 55 m wire long enough after allowing 1.5 m extra? If not, how much more is needed?
Read the worked answer
In the right triangle TFA, the angle at A is 30° and the side opposite it is TF = h ≈ 27.32 m. If TA is the straight-line wire length, sin 30° = h/TA, so TA = h/sin 30° = h ÷ 0.5 = 2h ≈ 54.64 m. The installation needs 1.5 m extra, so the required length is about 54.64 + 1.5 = 56.14 m. The 55 m wire is therefore too short. The shortage is about 56.14 - 55 = 1.14 m, so the supplier should provide at least about 56.2 m of wire, for example a 57 m length.