Solution :

Given, the angles of elevation of the top of a tower from two points distant s and t from its foot are 30° and 60°.

To prove : Height of the tower = √(st).

Let the height of the tower be h.

We know, tanθ = Perpendicular side/Adjacent side.

At first, from the diagram attached,

From ∆ABD,

→ tan60° = h/t

→ h/t = √3 ...(i)

Again, from ∆ABC,

→ tan30° = h/s

→ 1/√3 = h/s

→ h/s = 1/√3 ...(ii)

Now multiplying (i) and (ii) :

→ h/t × h/s = √3 × 1/√3

→ h²/st = 1

→ h² = st

→ h = √(st)

Therefore,

Height of the tower = √(st). [Proved]