2022 ASRJC P1 Q11

The cylindrical tank in a research laboratory has a cross-sectional area of \(4\) m\(^{2}\). To cool the tank, water is pumped in and out of the tank simultaneously. The volume and height of the water in the tank at any time \(t\) minutes is given by \(V\)(litres) and \(h\) (metres) respectively. Clean water is pumped into the tank at a rate that is proportional to \({{h}^{2}}\) and the water is pumped out from the tank at a rate that is proportional to \(h\).

1. Assume that the water does not overflow and that there is no change to the height of the water when \(h\) is \(10\), show that \(\frac{\mathrm{d}h}{\mathrm{d}t}=\frac{kh(h-10)}{4}\) where \(k\) is a real constant.[4]

The tank was initially filled with clean water to a height of \(2\) metres. When the height of the water is \(5\) metres, the volume of water is increasing at a rate of \(5.5\) litres per minute.

2. Find the exact value of \(k\). Hence find \(h\) in terms of \(t\).[5]
3. Sketch a graph of \(h\) against \(t\). Hence, write down the minimum height of the cylindrical tank that will not result in the overflow of the water.[3]