6. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} The shaded region, shown in Figure 4, is bounded by the \(x\)-axis, the line with equation \(x = 6\), the line with equation \(y = 2\) and the \(y\)-axis. This region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { x }\)-axis to form a solid of revolution. This solid is used to model a non-uniform cylinder of height 6 cm and radius 2 cm . The mass per unit volume of the cylinder at the point \(( x , y , z )\) is \(\lambda ( x + 2 ) \mathrm { kg } \mathrm { cm } ^ { - 3 }\), where \(0 \leqslant x \leqslant 6\) and \(\lambda\) is a constant.

1. Show that the mass of the cylinder is \(120 \lambda \pi \mathrm {~kg}\).
2. Show that the centre of mass of the cylinder is 3.6 cm from \(O\). The point \(O\) is the centre of one end of the cylinder. The point \(A\) is the centre of the other end of the cylinder. A uniform solid hemisphere of radius 3 cm has density \(\lambda \mathrm { kg } \mathrm { cm } ^ { - 3 }\). The hemisphere is attached to the cylinder with the centre of its circular face in contact with the point \(A\) on the cylinder to form the model shown in Figure 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_309_673_1713_696} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The model is placed with the end containing \(O\) on a rough inclined plane which is inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent the model from sliding. The model is on the point of toppling.
3. Find the value of \(\alpha\).