A lighthouse is \(65\) m tall. Two boats, \(P\) and \(Q\), are on the sea in a straight line with the foot of the lighthouse, with \(P\) further away than \...

An angle of depression is measured downwards from the horizontal at the observer's eye. Because the horizontal at the top of the lighthouse is parallel to the sea, each angle of depression equals the angle of elevation of the lighthouse top from the boat, by alternate angles. So each boat sits in a right-angled triangle whose vertical side is the lighthouse, 65 m, and whose horizontal side is that boat's distance from the foot.

For each boat the tangent ratio links the height to the horizontal distance, \(\tan(\text{angle})=\dfrac{65}{\text{distance}}\), so the distance is \(\dfrac{65}{\tan(\text{angle})}\).

• Boat \(P\): \(\dfrac{65}{\tan 24^\circ}=\dfrac{65}{0.44523\ldots}=146.0\ldots\) m. [M1] [A1]
• Boat \(Q\): \(\dfrac{65}{\tan 41^\circ}=\dfrac{65}{0.86929\ldots}=74.77\ldots\) m. [M1]

Both boats lie on the same side in a straight line with the foot of the lighthouse, so \(PQ\) is the difference of the two distances: \[PQ=146.0\ldots-74.77\ldots=71.2\text{ m}\] [A1]

The steeper angle of depression belongs to the nearer boat, which is why \(Q\) at \(41^\circ\) is much closer than \(P\) at \(24^\circ\). Multiplying by the tangent instead of dividing is the usual error and would place both boats absurdly close to the lighthouse. Keep the unrounded distances until the subtraction, since rounding each to 3 significant figures first can shift the final answer.