A uniform rod \(A B\) of length \(4 a\) and weight \(W\) is smoothly hinged to a vertical wall at the end \(A\). The rod is held at an angle \(\theta\) above the horizontal by a light elastic string. One end of the string is attached to the point \(C\) on the rod, where \(A C=3 a\). The other end of the string is attached to a point \(D\) on the wall, with \(D\) vertically above \(A\) and such that angle \(A C D=2 \theta\). A particle of weight \(\frac{1}{2} W\) is attached to the rod at \(B\). It is given that \(\tan \theta=\frac{8}{15}\).
(i) Show that the tension in the string is \(\frac{17}{12} W\).
(ii) Find the magnitude and direction of the reaction at the hinge.

(iii) Given that the natural length of the string is \(2 a\), find its modulus of elasticity.