2020 NYJC Promo Q11 [Modified]

[The formula for the volume of a cone of height \(h\) and base radius \(r\) is \(\frac{1}{3}\pi {{r}^{2}}h\) and the formula of the arc length of a sector with radius \(r\) and subtended at an angle \(\theta \) is \(r\theta \).]

PartyMoMo is a kid’s party planner company and they plan to design a party hat in the shape of a cone.

The designer uses a piece of circular paper with radius \(9\) units in Fig. 1, and he removes a minor sector to form the shape shown in Fig. 2. The sector remaining has an angle \(\theta \), in radians, subtended at the centre by the arc of the sector. The paper is now joined at the dotted lines to form the open cone in Fig. 3, which has a circular base with a circumference equal to the arc length in Fig. 2. The cone has slant height \(9\) units and the radius of the base is \(r\) units.

The vertical height \(h\) and the volume \(V\) of the party hat are given by \(h=\frac{9}{2\pi }\sqrt{4{{\pi }^{2}}-{{\theta }^{2}}}\) and \(\frac{243}{8{{\pi }^{2}}}{{\theta }^{2}}\sqrt{4{{\pi }^{2}}-{{\theta }^{2}}}\) as \(\theta \) varies.

1. Using differentiation, find the exact value of \(\theta \) that gives a stationary value of \(V\), explaining why there is only one answer. Justify that this value of \(\theta \) gives a maximum value for \(V\).[5]
2. The designer decides to use a 3D printer to print a scale model of the party hat by setting \(\theta =\frac{4}{3}\pi \) that always satisfies the equation \(V=\frac{243}{8{{\pi }^{2}}}{{\theta }^{2}}\sqrt{4{{\pi }^{2}}-{{\theta }^{2}}}\). Show that the height of the scale model of the party hat is \(3\sqrt{5}\) units.[1]

After printing the scale model, the designer fills the inverted party hat to its capacity with fine sand. Due to a hole in the 3D printing at the vertex of the inverted party hat, sand is leaking at a constant rate of \(3\) units\(^{3}\)s\(^{-1}\) when the inverted party hat is upright in position with the vertex below its circular base. Given that the radius of the base is \(6\) units, find the exact rate at which the height of the sand is decreasing at the instant when the height of the sand is \(5\) units.[3]