2003 HCI P2 Q2

Liquid is poured into a container at a constant rate of \(30\) cm\(^{3}\)s\(^{-1}\) and leaks out at a rate which is proportional to the volume of liquid in the container. At time \(t\) seconds, the volume of the water in the container is \(V\)cm\(^{3}\). When the volume reaches \(450\) cm\(^{3}\), it decreases at a rate of \(30\) cm\(^{3}\)s\(^{-1}\) .

Show that \(-15\frac{\mathrm{d}V}{\mathrm{d}t}=2V-450\).[2]

It is given that \(V=1000\) cm\(^{3}\) when \(t=0\).

1. Show that \(V=A{{e}^{\alpha \,t}}+B\), where \(\alpha \), \(A\) and \(B\) are constants to be determined.[4]
2. Sketch the graph of \(V\) against \(t\).[1]