(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{2pi 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}).