M4 June 2008 Q6

6.

A river is 30 m wide and flows between two straight parallel banks. At each point of the river, the direction of flow is parallel to the banks. At time \(t = 0\), a boat leaves a point \(O\) on one bank and moves in a straight line across the river to a point \(P\) on the opposite bank. Its path \(OP\) is perpendicular to both banks and \(OP = 30\) m, as shown in Figure 2. The speed of flow of the river, \(r\) m s\(^{-1}\), at a point on \(OP\) which is at a distance \(x\) m from \(O\), is modelled as\[r = \frac{1}{10}x,\qquad 0 \leqslant x \leqslant 30.\]

The speed of the boat relative to the water is constant at 5 m s\(^{-1}\). At time \(t\) seconds the boat is at a distance \(x\) m from \(O\) and is moving with speed \(v\) m s\(^{-1}\) in the direction \(OP\).