A2 June 2019 Q5

5.

The region \(R\), shown shaded in Figure 4, is bounded by part of the curve with equation \(y^2 = 2x\), the line with equation \(y = 2\) and the \(y\)-axis. The unit of length on both axes is one centimetre. A uniform solid, \(S\), is formed by rotating \(R\) through 360° about the \(y\)-axis.

Given that the volume of \(S\) is \(\dfrac{8}{5}\pi\ \text{cm}^3\),

A uniform solid cylinder, \(C\), has base radius 2 cm and height 4 cm. The cylinder \(C\) is attached to \(S\) so that the plane face of \(S\) coincides with a plane face of \(C\), to form the paperweight \(P\), shown in Figure 5. The density of the material used to make \(S\) is three times the density of the material used to make \(C\).

The plane face of \(P\) rests in equilibrium on a desk lid that is inclined at an angle \(\theta^\circ\) to the horizontal. The lid is sufficiently rough to prevent \(P\) from slipping. Given that \(P\) is on the point of toppling,