The angle of elevation of the top of a building from the foot of a tower is 30°. The angle of elevation of the top of the tower when seen from the foot of the building is 60°. If the tower is 60 m high, then the height of the building is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 20 m
The tower and the building stand on the same ground, a distance \(d\) apart. Each angle of elevation gives \(\tan\theta=\frac{\text{height}}{\text{distance}}\), and the common distance \(d\) links the two heights.
From the foot of the building, \(\displaystyle \tan60^\circ=\frac{60}{d}\), so \(\displaystyle d=\frac{60}{\sqrt{3}}=20\sqrt{3}\) m
From the foot of the tower, \(\displaystyle \tan30^\circ=\frac{h}{d}\), so \(\displaystyle h=20\sqrt{3}\times\frac{1}{\sqrt{3}}=20\) m
Shortcut: The distance is the same in both, so the heights are in the ratio of the tangents.
\(\displaystyle \frac{h}{60}=\frac{\tan30^\circ}{\tan60^\circ}=\frac{1/\sqrt{3}}{\sqrt{3}}=\frac{1}{3}\), so \(h=20\) m