Question 7 - A-Level Maths

\hspace{0pt} [You may assume that the volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).] A uniform solid right circular cone \(C\), with vertex \(V\), has base radius \(r\) and height \(h\).
1. Show that the centre of mass of \(C\) is \(\frac { 3 } { 4 } h\) from \(V\)
A solid \(F\), shown below in Figure 4, is formed by removing the solid right circular cone \(C ^ { \prime }\) from \(C\), where cone \(C ^ { \prime }\) has height \(\frac { 1 } { 3 } h\) and vertex \(V\) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-24_666_670_854_639} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure}
Show that the distance of the centre of mass of \(F\) from its larger plane face is \(\frac { 3 } { 13 } h\) The solid \(F\) rests in equilibrium with its curved surface in contact with a horizontal plane.
Show that \(13 r ^ { 2 } \leqslant 17 h ^ { 2 }\)

\includegraphics[max width=\textwidth, alt={}]{9777abb8-a564-40d5-8d96-d5649913737b-28_2642_1844_116_114}