Question : Let A and B be two towers with the same base. From the midpoint of the line joining their feet, the angles of elevation of the tops of A and B are 30° and 60°, respectively. The ratio of the heights of B and A is:
Option 1: 1 : 3
Option 2: 3 : 1
Option 3: 1 : 2
Option 4: $1: \sqrt{3}$
Correct Answer: 3 : 1
Solution :
There are two towers A and B
The angles of elevation of the tops of A and B are 30° and 60°, respectively
$\angle$ AOC = 30°, $\angle$ BOD = 60°
OC = OD
$\tan \theta = \frac{\text{Perpendicular}}{\text{Base}}$
Let AC = h
Then, BD = H
In triangle AOC,
$\tan 30^\circ = \frac{\text{h}}{\text{OC}}$
$\frac{1}{\sqrt{3}} = \frac{\text{h}}{\text{OC}}$ -----(1)
In triangle BOD,
$\tan 60^\circ = \frac{\text{H}}{\text{OD}}$
$\frac{\sqrt{3}}{1} = \frac{\text{H}}{\text{OD}}$ -----(2)
By Dividing eq (1) and eq (2)
$\frac{1}{3} = \frac{\text{h}}{\text{H}}$
⇒ H : h = 3 : 1
Then the ratio of the height of tower B to tower A = 3 : 1
Hence, the correct answer is 3 : 1.
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