A swimmer crosses a flowing stream of width d' to and fro in time `t_(1)`. The time taken to cover the same distance up and down the stream is `t_(2)`, If `t_(3)` is the time the swimmer would take to swim a distance 2d in still water, then

Explanation

Let v be the river velocity and u the velocity of swimmer in still water. Then
`t_(2)=2((d)/(sqrt(u^(2)-v^(2))))" "....(i)`
`t_(2)=(d)/(u+v)+(d)/(u-v)=(2ud)/(u^(2)-v^(2))" "...(ii) and t_(3)=(2d)/(u) " "....(iii)`
from equations (i), (ii) and (iii)
`t_(1)^(2)=t_(2)t_(3) rArr = sqrt(t_(2)t_(3))`