WAEC 2022 · Paper 2 · Q10

Elevation, depression & bearings
Trigonometric ratios

Two canoes $M$ and $N$ started from a shore $A$ at the same time. $M$ sails at $35\text{ km/h}$ on a bearing of $230^\circ$ while $N$ sails at $30\text{ km/h}$ on a bearing of $320^\circ$ . If the two canoes sailed for 2.5 hours, find, correct to one decimal place, the:

(a)
distance between $M$ and $N$ ;
(b)
bearing of $M$ from $N$ .

Worked solution (try it first)

In 2.5 hours, $M$ sails $35 \times 2.5 = 87.5$ km on $230^\circ$ and $N$ sails $30 \times 2.5 = 75$ km on $320^\circ$ .
The bearings differ by $320^\circ - 230^\circ = 90^\circ$ , so the triangle $AMN$ is right-angled at $A$ .

(a)

$|MN| = \sqrt{75^2 + 87.5^2}$
$= \sqrt{13\,281.25}$
$\approx 115.2$ km.

(b)

At $N$ : $\tan\angle ANM = \frac{87.5}{75}$ , so $\angle ANM \approx 49.4^\circ$ .
At $N$ , the direction back to $A$ is $320^\circ - 180^\circ = 140^\circ$ , and $M$ is $49.4^\circ$ further round clockwise.
Bearing of $M$ from $N$ $= 140^\circ + 49.4^\circ = 189.4^\circ$ .

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