A uniform $$\(\operatorname{rod} A B\)$$ has length $$\(8 a\)$$ and weight $$\(W\)$$. The end $$\(A\)$$ of the rod is freely hinged to horizontal ground. The rod rests in equilibrium against a block which is also fixed to the ground. The block is modelled as a smooth solid hemisphere with radius $$\(2 a\)$$ and centre $$\(D\)$$. The point of contact between the rod and the block is $$\(C\)$$, where $$\(A C=5 a\)$$ The rod is at an angle $$\(\theta\)$$ to the ground, as shown in Figure 1. Points $$\(A, B, C\)$$ and $$\(D\)$$ all lie in the same vertical plane. (a) Show that $$\(A D=\sqrt{29} a\)$$ (1) (b) Show that the magnitude of the normal reaction at $$\(C\)$$ between the rod and the block is $$\(\frac{4}{\sqrt{29}} W\)$$ (3) The resultant force acting on the rod at $$\(A\)$$ has magnitude $$\(k W\)$$ and acts at angle $$\(\alpha\)$$ to the ground. (c) Find (i) the exact value of $$\(k\)$$ (ii) the exact value of $$\(\tan \alpha\)$$ (8)

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