(a) Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius $$\(r\)$$ is at a distance $$\(\frac{3}{8} r\)$$ from the centre of its plane face. [You may assume that the volume of a sphere of radius $$\(r\)$$ is $$\(\frac{4}{3} \pi r^{3}\)$$ ] (5) A uniform solid hemisphere of radius $$\(r\)$$ is joined to a uniform solid right circular cone made of the same material to form a toy. The cone has base radius $$\(r\)$$ and height $$\(k r\)$$. The centre of the base of the cone is $$\(O\)$$. The plane face of the cone coincides with the plane face of the hemisphere, as shown in Figure 3. The toy can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane. (b) Find the exact value of $$\(k\)$$ (5)

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