WAEC 2009 · Paper 2 · Q13✱

(b)
The pilot of an aircraft 2000 metres above sea level observes at an instant that the angles of depression of two boats, which are in a direct straight line with the aircraft, are $58^\circ$ and $72^\circ$ . Find, correct to the nearest metre, the distance between the two boats.

Worked solution (try it first)

Let $X$ be the point on the sea directly below the aircraft $P$ , with $|PX| = 2000$ m, and let $M$ and $N$ be the nearer and farther boats.
The angle of depression of $M$ is $72^\circ$ , so $\angle XPM = 90^\circ - 72^\circ$
$= 18^\circ$ .
For $N$ , $\angle XPN = 90^\circ - 58^\circ$
$= 32^\circ$ .
$|XM| = 2000\tan 18^\circ = 649.84$ m and $|XN| = 2000\tan 32^\circ = 1249.74$ m.
Distance between the boats $= 1249.74 - 649.84 = 599.9$ m, which is 600 m to the nearest metre.