WAEC 2018 · Paper 2 · Q9

(a)
A donkey is tied with a rope to a post which is $15\text{ m}$ from a fence. If the length of the rope between the donkey and the post is $17\text{ m}$ , calculate the length of the fence within the reach of the donkey.
(b)
The base of a right pyramid with vertex $V$ is a square $PQRS$ of side $15\text{ cm}$ . If the slant edge is $32\text{ cm}$ long, (i) represent the information in a diagram; (ii) calculate its: (I) height, correct to one decimal place; (II) volume, correct to the nearest $\text{cm}^3$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

Draw the perpendicular from the post to the fence (15 m).
The rope (17 m) reaches the fence at two points, each the hypotenuse of a right-angled triangle: $x = \sqrt{17^2 - 15^2} = \sqrt{64} = 8$ m on each side of the foot.
The donkey reaches $2 \times 8 = 16$ m of fence.

Draw the square base $PQRS$ with its diagonals crossing at the centre $M$ , and the vertex $V$ directly above $M$ , with the slant edges $VP$ , $VQ$ , $VR$ , $VS$ of 32 cm.

(I)** The diagonal of the base is $15\sqrt2 \approx 21.21$ cm, so $MP \approx 10.61$ cm.
In the right-angled triangle $VMP$ : height $VM = \sqrt{32^2 - 10.61^2}$
$= \sqrt{1024 - 112.5}$
$\approx 30.2$ cm.
(II) Volume $= \frac13 \times 15^2 \times 30.19$
$\approx 2264\text{ cm}^3$ .