A boat can be rowed at 5 m s^-1 in still water across a river of width 200 m flowing at 3 m s^-1. The time taken to cross by the path of shortest…

Least time: head perpendicular to the banks, so the time is $\dfrac{200}{5}=40\ \mathrm{s}$ (the current only carries the boat downstream by $120\ \mathrm{m}$). Shortest path: the boat must point upstream at angle $\theta$ with $\sin\theta=\dfrac35$ so that the downstream drift cancels; the cross-stream speed is then $\sqrt{5^2-3^2}=4\ \mathrm{m\,s^{-1}}$ and the time is $\dfrac{200}{4}=50\ \mathrm{s}$. The difference is $50-40=10\ \mathrm{s}$. Using $5\ \mathrm{m\,s^{-1}}$ as cross speed in both cases would give zero difference.