From the top of a tower, the angles of depression of two objects A and B (situated on the ground on the same side of the tower) are observed to be 30° and 60°, respectively. If the distance between the objects is $200\sqrt{3} m$, then the height of the tower is?

$300\, m$

The correct answer is Option (4) → $300\, m$

Step 1: Let the height of the tower be h

The object with angle of depression 60° is nearer to the tower,
and the object with angle of depression 30° is farther.

Step 2: Find horizontal distances

For nearer object:

$\tan 60^\circ = \frac{h}{x} \Rightarrow x = \frac{h}{\sqrt{3}}$

For farther object:

$\tan 30^\circ = \frac{h}{y} \Rightarrow y = h\sqrt{3}$

Step 3: Distance between the objects

$y - x = h\sqrt{3} - \frac{h}{\sqrt{3}} = \frac{2h}{\sqrt{3}}$

Given:

$\frac{2h}{\sqrt{3}} = 200\sqrt{3}$

Step 4: Solve for h

$2h = 600$

$h = 300$