A girl standing on level ground sees the top of a tower at an angle of elevation of \(30^\circ\). After she walks 50 m straight towards the foot of the tower, the angle of elevation becomes \(45^\circ\). Ignoring her height, find the height of the tower.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(25(\sqrt{3}+1)\) m
Each position of the girl gives a right-angled triangle with the tower as the side opposite the angle of elevation, so \(\text{height}=\text{distance}\times\tan(\text{angle})\). Writing the height from both positions gives one equation for the unknown distance.
Let the second position be \(x\) m from the foot; the first position is \((x+50)\) m from it
Height \(h=x\tan 45^\circ=x\)
Height \(h=(x+50)\tan 30^\circ=\displaystyle\frac{x+50}{\sqrt{3}}\)
\(\sqrt{3}\,x=x+50\), so \(x=\displaystyle\frac{50}{\sqrt{3}-1}=\frac{50(\sqrt{3}+1)}{3-1}=25(\sqrt{3}+1)\)
\(h=x=25(\sqrt{3}+1)\) m
