WAEC 2008 · Paper 2 · Q1

(a)
If $x : y = 2 : 3$ , evaluate $\dfrac{x^2 - y^2}{y^2 + x^2}$ .
(b)
A man on the same level ground with a tree stands at a distance of $12.82\text{ m}$ from the foot of the tree. He observes the angle of elevation of the top of the tree as $52^\circ$ . If the man is $1.24\text{ m}$ tall, calculate, correct to two decimal places, the height of the tree.

Worked solution (try it first)

(a)

A ratio of $2 : 3$ means $x = 2k$ and $y = 3k$ for some number $k$ .
Put these in: $x^2 - y^2 = 4k^2 - 9k^2 = -5k^2$ and $y^2 + x^2 = 9k^2 + 4k^2 = 13k^2$ .
Divide, and $k^2$ cancels: the value is $-\frac{5}{13}$ .

(b)

Draw the right-angled triangle from the man's eye: the horizontal side is $12.82\text{ m}$ and the angle at his eye is $52^\circ$ .
The height of the tree above his eye is $h = 12.82 \tan 52^\circ$
$= 12.82 \times 1.2799$
$= 16.409\text{ m}$ .
Add the man's height, because the angle is measured from his eye: $16.409 + 1.24 = 17.649$ .
The tree is $17.65\text{ m}$ tall, to two decimal places.

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