# A boat, which has a speed of 5km/h in still water, crosses a river of width 1km along the shortest path possible in 15 minutes. Find the velocity of the river?

Jun 7, 2018

The river is moving at a speed of 3 km/h.

#### Explanation:

If the speed of the river is constant over the stretch that the boat is sailing, we can use Pythagoras:

If the river had been still, the boat would have moved from A to B, 1 km.
Instead, because the water is moving, the boat moves from A to C,
This stretch must be $\frac{5}{4} k m = 1.25 k m$, since 15 min. = $\frac{1}{4}$ hour, in which time the boat moves $\frac{5}{4} k m$.

According to Pythagoras,
$C {B}^{2} {=}^{A} {C}^{2} - A {B}^{2}$
$= {\left(\frac{5}{4}\right)}^{2} - {1}^{2}$
$= \frac{25}{16} - 1 = \frac{9}{16}$
Therefore $B C = \sqrt{\frac{9}{16}} = \frac{3}{4}$

As the river, therefore, moves $\frac{3}{4} k m$ in $\frac{1}{4}$ hour, it follows that the speed is 3 km/h