From a point exactly midway between the foot of two towers P and Q, the angle of elevation of their tops are 30° and 60°, respectively. The ratio of the heights of tower P to that of Q is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1 : 3
Each tower, the ground and the line of sight form a right triangle, in which \(\tan\theta=\frac{\text{height}}{\text{distance}}\). The observer is midway, so both towers are the same distance \(d\) away and only the angles differ.
Tower P: \(\displaystyle h=d\tan30^\circ=\frac{d}{\sqrt{3}}\)
Tower Q: \(H=d\tan60^\circ=\sqrt{3}\,d\)
\(\displaystyle \frac{h}{H}=\frac{d/\sqrt{3}}{\sqrt{3}\,d}=\frac{1}{3}\), so the ratio is 1 : 3