WAEC 2016 · Paper 2 · Q11

(a)
If $\dfrac{3p + 4q}{3p - 4q} = 2$ , find $p : q$ .
Show the answer
$4 : 1$
(b)
The diagram shows the cross section $PQRSTU$ of a bridge with a semicircular hollow $TS$ in the middle. $|PU| = |QR| = 4\text{ m}$ and $|UT| = |SR| = 2\text{ m}$ . If the perimeter of the cross section is $34\text{ m}$ , calculate the: (i) length $PQ$ ; (ii) area of the cross section. $\left[\text{Take }\pi = \frac{22}{7}\right]$
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

Let the semicircle have radius $r$ .
The width across the bottom is $2 + 2r + 2$ , so $|PQ| = 4 + 2r$ .
The perimeter goes round $PQ$ , down $QR$ (4), along $RS$ (2), round the semicircle ( $\frac{22}{7}r$ ), along $TU$ (2) and up $UP$ (4): $(4 + 2r) + 4 + 2 + \frac{22}{7}r + 2 + 4 = 34$ .
So $16 + \frac{36}{7}r = 34$ , $\frac{36}{7}r = 18$ and $r = 3.5$ m.
Then $|PQ| = 4 + 7 = 11$ m.

Area $=$ rectangle $-$ semicircle $= 11 \times 4 - \frac12 \times \frac{22}{7} \times 3.5^2$
$= 44 - 19.25$
$= 24.75\text{ m}^2$ .