(a) If \(\frac{3p + 4q}{3p – 4q} = 2\), find \(p : q\).
(b) 
The diagram shows the cross section of a bridge with a semi-circular hollow in the middle. If the perimeter of the cross section is 34 cm, calculate the :
(i) length PQ; (ii) area of the cross section.
[Take \(\pi = \frac{22}{7}\)].
Explanation
(a) \(\frac{3p + 4q}{3p - 4q} = 2\)
\(3p + 4q = 2(3p - 4q)\)
\(3p + 4q = 6p - 8q\)
\(3p - 6p = - 8q - 4q\)
\(-3p = - 12q\)
\(p = 4q\)
\(\frac{p}{q} = \frac{4}{1}\)
\(\therefore p : q = 4 : 1\).
(b)(i) 
Perimeter of cross- section = 4 + x + 4 + 2 + y + 2
i.e 34 = 12 + x + y
x + y = 22 ..... (1)
From the diagram, |PQ| = |UR| (opp. sides of a rectangle)
i.e. x = 2 + d + 2
x - 4 = d ..... (2)
From (1), y = 22 - x is the circumference of the semi-circle.
\(22 - x = \frac{2\pi r}{2} = \pi r\)
\(r = \frac{d}{2} \)
\(22 - x = \frac{22}{7} \times \frac{(x - 4)}{2}\)
\(154 - 7x = 11x - 44\)
\(154 + 44 = 11x + 7x \)
\(198 = 18x\)
\(x = \frac{198}{18} = 11 m\)
Hence, |PQ| = 11 m.
(ii) Area of cross section = Area of rectangle PQRU - area of semi-circle
Area of rectangle = \(11 \times 4 = 44 m^{2}\)
Area of semi-circle = \(\frac{\pi r^{2}}{2} \)
= \(\frac{22}{7} \times (\frac{(11 - 4)}{2})^{2} \times \frac{1}{2}\)
= \(19.25 m^{2}\)
Area of cross section = \(44 - 19.25 = 24.75 m^{2}\).