The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 50/3 m
The tower and the building stand on the same ground, a distance \(d\) apart. Each angle of elevation makes a right triangle with this distance, and \(\tan(\text{angle})=\text{height}\div d\); the tower’s triangle gives \(d\), and \(d\) then gives the building’s height \(h\).
From the foot of the building, the top of the 50 m tower is at \(60^\circ\): \(\displaystyle \tan60^\circ=\frac{50}{d}\), so the distance \(\displaystyle d=\frac{50}{\sqrt{3}}\) m (1)
From the foot of the tower, the top of the building is at \(30^\circ\): \(\displaystyle \tan30^\circ=\frac{h}{d}\), so the height \(\displaystyle h=\frac{d}{\sqrt{3}}\) (2)
Putting \(d\) from (1) into (2), the height of the building \(\displaystyle h=\frac{50}{\sqrt{3}}\times\frac{1}{\sqrt{3}}=\frac{50}{3}\) m