2019 HCI P1 Q11

A container is made up of a cylinder and an inverted right circular cone as shown in the diagram below. The height and the diameter of the cylinder are \(20\) cm and \(5\) cm respectively. The height of the cone is \(4\) cm. An external device ensures liquid flows out through a small hole at the vertex of the cone into a bowl below at a constant rate of \(18\) cm\(^{3}\) per minute. The depth of the liquid and the radius of the liquid surface area in the container at the time \(t\) minutes are \(x\) cm and \(~r\) cm respectively. The container is full of liquid initially.

1. When \(x>4\), find the rate of change of the depth of the liquid in the container.[4]
2. Find the rate of change of \(r\) when \(x=2\).[5]
3. The bowl as shown in the diagram in part (a) is generated by rotating part of the curve \(\frac{{{x}^{2}}}{225}+\frac{{{y}^{2}}}{100}=1\) which is below the \(x\)-axis through \(\pi \) radians about the \(y\)-axis. Assuming the bowl has negligible thickness, find the volume of the empty space in the bowl when the liquid has completely flowed from the container into the bowl, giving your answer correct to \(3\) decimal places.[3]