WAEC 2024 · Paper 2 · Q8

Elevation, depression & bearings

Two observers, Abu and Badu, $46\text{ m}$ apart, observe a bird on top of a vertical pole from the same side of the pole. The angles of elevation of the bird from Abu and Badu are $40^\circ$ and $48^\circ$ respectively. Abu, Badu and the foot of the pole are on the same horizontal line.

(a)
Illustrate the information in a diagram.
Model answer
A clear sketch is enough (it need not be to scale), but it must show every given fact: draw the vertical pole $FB$ , with the bird at the top $B$ and a right angle at the foot $F$ . Both observers are on the same side of the pole: Badu ( $D$ ), nearer, sees the bird at $48^\circ$ , and Abu ( $A$ ), 46 m further back, sees it at $40^\circ$ . Let $|DF| = x$ and the height be $h$ .
(b)
Calculate, correct to one decimal place, the height of the pole (m).

Worked solution (try it first)

(a)

Draw the pole $FB$ upright with the bird at the top $B$ .
Badu $D$ is nearer the pole (the larger angle, $48^\circ$ ) and Abu $A$ is 46 m further back on the same line (the smaller angle, $40^\circ$ ).

(b)

Let Badu be $x$ m from the foot.
From each observer: $h = x\tan 48^\circ$ and $h = (x + 46)\tan 40^\circ$ .
Set them equal: $x(\tan 48^\circ - \tan 40^\circ) = 46\tan 40^\circ$ .
So $x = \frac{46 \times 0.8391}{1.1106 - 0.8391}$
$= \frac{38.60}{0.2715}$
$\approx 142.2$ m.
Then $h = 142.2 \times 1.1106 \approx 157.9$ m.

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