A boat sails in a straight line towards a lighthouse that stands at the water's edge. From the boat, the top of the lighthouse is first seen at an angle of elevation of \(45^\circ\); after the boat has covered another 40 m, the angle is \(60^\circ\). Treating the boat as a point at water level, find the height of the lighthouse.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(20(3+\sqrt{3})\) m
Each position of the boat gives a right-angled triangle with the lighthouse 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 \((x+40)\) m from it
Height \(h=x\tan 60^\circ=\sqrt{3}\,x\)
Height \(h=(x+40)\tan 45^\circ=x+40\)
\(\sqrt{3}\,x=x+40\), so \(x=\displaystyle\frac{40}{\sqrt{3}-1}=\frac{40(\sqrt{3}+1)}{3-1}=20(\sqrt{3}+1)\)
\(h=\sqrt{3}\times 20(\sqrt{3}+1)=20(3+\sqrt{3})\) m
