A flagstaff stands on the top of a vertical tower. The angle of elevation of the top of the flagstaff from a certain place on the same horizontal level with the base of the tower is found to be α. Advancing a distance d towards the tower in the same horizontal plane, the angle of elevation of the top of the flagstaff is observed to be β and that of the top of the tower is observed to be γ. Let H be the height of the top of the flagstaff from the base of the tower and h be the height of the tower. Which one of the following is correct?
- Ad = H(cot α − cot β)/2
- Bd = H(tan α − tan β)/2
- Cd = H(cot α − cot β)
- Dd = H(tan α − tan β)
Explanation
Let the tower base be O and the flagstaff top be T, with OT = H above ground. From the first observation point P at distance x from O, the angle of elevation of T is α, so tan α = H/x, i.e. x = H·cot α. After advancing distance d towards the tower the observer is at distance x − d, where the flagstaff top now subtends β, so tan β = H/(x − d), i.e. x − d = H·cot β. Subtracting the two distance expressions gives d = x − (x − d) = H(cot α − cot β), which is option (C).