WAEC 2022 · Paper 2 · Q13

(a)
The diameter of a cylinder closed at both ends is $7\text{ cm}$ . If the total surface area is $209\text{ cm}^2$ , calculate the height. $\left[\text{Take }\pi = \frac{22}{7}\right]$
(b)
The points $X$ and $Y$ , $19\text{ m}$ apart, are on the same side of a tree. The angles of elevation of the top, $T$ , of the tree from $X$ and $Y$ on the horizontal ground with the foot of the tree are $43^\circ$ and $38^\circ$ respectively. (i) Illustrate the information in a diagram. (ii) Find, correct to one decimal place, the height of the tree.

Worked solution (try it first)

A closed cylinder has two circular ends and a curved side: total surface area $= 2\pi r^2 + 2\pi rh$ .
With $r = 3.5$ : $2 \times \frac{22}{7} \times 3.5^2 + 2 \times \frac{22}{7} \times 3.5 \times h = 209$ .
So $77 + 22h = 209$ , $22h = 132$ and $h = 6$ cm.

Draw the tree upright with $X$ and $Y$ on the same side of it, $Y$ 19 m further away.
The angle of elevation is $43^\circ$ from $X$ (nearer) and $38^\circ$ from $Y$ (farther).

Let $X$ be $y$ m from the foot.
Then $h = y\tan 43^\circ$ and $h = (y + 19)\tan 38^\circ$ .
Set them equal: $y(\tan 43^\circ - \tan 38^\circ) = 19\tan 38^\circ$ , so $y = \frac{19 \times 0.7813}{0.9325 - 0.7813}$
$\approx 98.2$ m.
So $h = 98.2 \times 0.9325 \approx 91.5$ m.
(Rounding the tangents early can move the last digit.)