Tower, Flagstaff and Two Observation Points
A surveyor stands at point P on level ground and measures the angle of elevation of the top of a vertical tower as 45°. Walking 20 m straight towards the foot of the tower to point Q, the angle of elevation becomes 60°. On top of the tower stands a vertical flagstaff. From P, the flagstaff subtends an angle of 15°. The surveyor must find the height of the tower, the horizontal distance from P to the foot of the tower, and the length of the flagstaff. He also has only a 30 m tape and wants to know whether it can measure the horizontal distance from P to the foot. Use √3 ≈ 1.732, keep all points in one vertical plane, and assume the ground is horizontal and the tower is vertical. The flagstaff is not assumed to be equal to the tower height.
1. Find the height of the tower and the horizontal distance from P to the foot of the tower.
Read the worked answer
Let h be tower height and x the distance from P to the foot. From P: h/x = tan 45° = 1, so h = x. From Q, which is 20 m closer, h/(x − 20) = tan 60° = √3.
Substitute h = x: x = √3(x − 20), so x = √3 x − 20√3. Thus x(√3 − 1) = 20√3, giving x = 20√3/(√3 − 1). Rationalise: x = 20√3(√3 + 1)/2 = 10(3 + √3) ≈ 10(4.732) = 47.32 m.
So the tower height is also about 47.32 m, and the distance from P to the foot is about 47.32 m.
2. Find the length of the flagstaff.
Read the worked answer
From P, the elevation of the tower top is 45° and the flagstaff subtends 15°, so the elevation of the top of the flagstaff is 45° + 15° = 60°.
Let H be the height of the flagstaff top above ground. Then H/x = tan 60° = √3, so H = x√3 = (30 + 10√3)√3 = 30√3 + 30 ≈ 81.96 m.
Flagstaff length = H − h = (30√3 + 30) − (30 + 10√3) = 20√3 ≈ 34.64 m.
3. Decide whether the 30 m tape can measure the horizontal distance from P to the foot of the tower.
Read the worked answer
The required horizontal distance is x ≈ 47.32 m. A 30 m tape is shorter than 47.32 m, so it cannot measure the distance in one stretch. The surveyor would need to measure in at least two segments or use a longer tape. The tape is sufficient only if the distance were ≤ 30 m, which is not the case.