N2001 P2 Q14

A rectangular tank has a horizontal base. Water is flowing into the tank at a constant rate, and 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.

If the depth is \(0.5\)m, it remains at this constant value.

Show that \(\frac{\mathrm{d}x}{\mathrm{d}t}=-k\left( 2x-1 \right)\), where \(k\) is a positive constant.

When \(t=0\), the depth of water in the tank is \(0.75\)m and is decreasing at a rate of \(0.01\) ms\(^{-1}\).

Find the time at which the depth of water is \(0.55\)m.