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A uniform rod \(AB\) of length \(4L \) and weight \(W\) is inclined at an angle \(\theta\) to the horizontal. Its lower end \(A\) rests on a fixed support and the rod is held in equilibrium by a string attached to the rod at a point \(C\) which is \(3L \) from \(A\). The reaction of the support on the rod acts in a direction \(\alpha\) to \(AC\) and the string is inclined at an angle \(\beta\) to \(CA\). Show that \[ \cot\alpha = 3\tan \theta + 2 \cot \beta\,. \] Given that \(\theta =30^\circ\) and \(\beta = 45^\circ\), show that \(\alpha= 15^\circ\).