Reading a Tower's Height from Two Angles of Elevation
A survey team must report the height of a mobile tower standing on level ground, without climbing it. From a point A on the ground, the angle of elevation of the top of the tower is measured as 30°. The team then walks 24 m in a straight line towards the foot of the tower, to a point B, and measures the angle of elevation as 60°. Both readings are taken with the theodolite at the same height, so the instrument height cancels out and can be ignored. The junior surveyor writes in the field book: 'Let the height be h. Then h/tan 30° − h/tan 60° = 24, so h = 24√3 ≈ 41.6 m.' The senior surveyor is not convinced; she also wants to know the horizontal distance from B to the foot of the tower, and whether the angle of elevation measured from a point 60 m from the foot would be smaller than 15°. (Use √3 ≈ 1.732, tan 15° ≈ 0.268.)
1. Find the correct height of the tower and identify exactly where the junior surveyor went wrong.
Read the worked answer
Let F be the foot of the tower, h = height, FB = d.
From B: tan 60° = h/d → h = d√3 …(1)
From A: AF = d + 24 and tan 30° = h/(d + 24) → h = (d + 24)/√3 …(2)
Equating: d√3 = (d + 24)/√3 → 3d = d + 24 → d = 12 m, so h = 12√3 ≈ 12 × 1.732 = 20.78 m ≈ 20.8 m.
The junior's equation is right: h/tan 30° − h/tan 60° = h√3 − h/√3 = 24. His algebra is wrong: h√3 − h/√3 = h(3 − 1)/√3 = 2h/√3 = 24, giving h = 12√3, not 24√3. He multiplied by √3 where he should have divided, so 41.6 m is exactly double the true height.
Quick check: from A the distance is 36 m and 36/√3 = 20.78 m ✔
2. Find the horizontal distance from B to the foot of the tower and show that the two readings are consistent.
Read the worked answer
From the working above, FB = d = 12 m and AF = 12 + 24 = 36 m.
Check the two angles with h = 20.78 m:
tan (angle at B) = 20.78/12 = 1.732 = tan 60° ✔
tan (angle at A) = 20.78/36 = 0.577 = tan 30° ✔
The readings are consistent with each other, which is why the height can be quoted as 20.8 m. Had the junior's 41.6 m been reported, the same field data would have demanded tan 60° = 41.6/12 = 3.47, an angle of about 74°, contradicting his own instrument reading - a useful consistency check to apply before signing a survey sheet.
3. Would the angle of elevation from a point 60 m from the foot be smaller than 15°? Justify with calculation.
Read the worked answer
At 60 m from the foot, tan θ = h/60 = 12√3/60 = √3/5 = 1.732/5 = 0.3464.
Given tan 15° ≈ 0.268. For 0° < θ < 90°, tan θ increases with θ, so 0.3464 > 0.268 implies θ > 15° (in fact θ ≈ 19.1°).
Verdict: the suggestion is false - the angle is larger than 15°.
Consistency check: 60 m is farther than A's 36 m, so the angle must be smaller than the 30° measured at A but need not be as small as 15°; 19.1° fits both conditions.