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?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 300 m
The angle of depression from the top of the tower equals the angle of elevation of the top as seen from the object, because the two horizontal lines are parallel. The nearer object B is seen at the larger angle, \(60^\circ\). In each right triangle with the tower, \(\tan\theta=\frac{\text{height}}{\text{distance}}\).
Equation (1) (from B):
\(\tan60^\circ=\displaystyle\frac{h}{x}\)
So \(\displaystyle x=\frac{h}{\sqrt{3}}\) (1)
Equation (2) (from A):
\(\tan30^\circ=\displaystyle\frac{h}{x+200\sqrt{3}}\)
So \(x+200\sqrt{3}=h\sqrt{3}\) (2)
Finding \(h\) (putting \(x\) from (1) into (2)):
\(\displaystyle \frac{h}{\sqrt{3}}+200\sqrt{3}=h\sqrt{3}\)
So \(\displaystyle h\sqrt{3}-\frac{h}{\sqrt{3}}=\frac{2h}{\sqrt{3}}=200\sqrt{3}\)
\(2h=200\times3=600\)
\(h=300\) m