2025 SST PRELIMS P1 Q11

A spherical scoop of ice cream with radius \(x\) cm rests on top of an inverted cone with height 18 cm and radius 7.5 cm.

As the ice cream scoop melts, the radius of the ice cream scoop decreases at a constant rate of 0.00167 cm/s.

1. Assuming that the scoop of ice cream retains its shape, find the rate at which the volume of the ice cream scoop is decreasing at the instant when \(x=8\).[3]
2. Show that the volume of the melted ice cream in the cone, \(C\), is \(C=\dfrac{25\pi}{432}h^3\), where \(h\) cm is the depth of the melted ice cream in the cone.[1]
3. The cone was observed to be 12% filled at some point in time. It is given that the rate of increase of the volume of melted ice cream in the cone is the same rate as the rate of decrease of the volume of the ice cream found in part (i). Assuming no melted ice cream drips outside the cone, find the rate of change of the depth of the melted ice cream in the cone at that instant.[4]