2022 ASRJC J2 MYCT P1 Q10

In a science research centre, a hollow truncated spherical force field shelter is created to act as an outer protective layer to an expansible cone-shaped vault used to store radioactive materials. The structure consists of a right circular cone of height \(h\) units, radius \(r\) units and slant height \(L\) units inscribed in the sphere of radius \(3\) units. Point \(C\) is the centre of the sphere and point \(P\) is a variable point on the spherical surface such that \(CP\) makes an angle of \(\theta \) radians with the horizontal (as shown in Figure 1).

1. Show that the curved surface area of the cone is given by\(A=9\pi \cos \theta \sqrt{2\sin \theta +2}\).[3]
2. Find the height \(h\) that gives the greatest value of \(A\).[4]
3. The volume of the cone-shaped vault is increasing at a rate of \(6\) units\(^{3}\)/s when \(\theta =\frac{\pi }{6}\). Determine the rate of change of \(h\) at this instant, leaving your answer in exact form.[5]

[It is given that the curved surface area of a right circular cone with base radius \(r\) and slant height \(l\) is \(\pi rl\).]