(a) If \(\frac{3p + 4q}{3p - 4q} = 2\), find \(p : q\). (b) The

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 = \(44 - 19.25 = 24.75 m^{2}\).