2015 TJC P2 Q3 (b)

An empty rectangular tank has vertical sides of depth \(H\) metres and a horizontal base of unit area. Water is pumped into the tank at a constant rate such that if no water flows out, the tank can be filled up in time \(T\) seconds. Water flows out at a rate which is proportional to the depth of water in the tank. At time \(t\) seconds, the depth of water in the tank is \(x\) metres.

When the depth of water is 1 metre, it remains at this constant value.Show that

\(\frac{\mathrm{d}x}{\mathrm{d}t}=k(1-x)\), where \(k\) is a constant in terms of \(H\) and \(T\).[2]

Find \(x\) in terms of \(t\), \(H\) and \(T\).[4]