2. The triangular region shown below is rotated through \(360 ^ { \circ }\) around the \(x\)-axis, to form a solid cone.
\includegraphics[max width=\textwidth, alt={}, center]{bab18666-3571-4906-ab2f-b85e87c6e8f4-3_316_716_479_671} The coordinates of the vertices of the triangle are \(( 0,0 ) , ( 8,0 )\) and \(( 0,4 )\).
All units are in centimetres.

1. State an assumption that you should make about the cone in order to find the position of its centre of mass.
   [0pt] [1 mark]
2. Using integration, prove that the centre of mass of the cone is 2 cm from its plane face.
   [0pt] [5 marks]
3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding.
4. 1. Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree.
      [0pt] [2 marks]
5. (ii) Find the range of possible values for the coefficient of friction between the cone and the board.