A man can row a boat with 4 km/h in still water. If the is crosssing a river where the current is 2km/h.
a In what direction will his boat be headed it he wants to reach a point on the other bank , directly opposite to starting point?
b.If width of the river is 4 km, how ling will the man take to cross the river, with the condition in part (a)?
c. In what direction should he head the boat if he wants to cross the river in shortest time and what is this minimum time?
d.how long will it take to row 2 km up the strem and then back to his starting point?

Explanation

a. Given that `v_(br)=4` km/h and `v_(r)=2` km/hr
`:.theta=sin^(-1)((v_(r))/(v_(bc)))`
`=sin^(-1)(2/4)=30^(@)`
Hence to reach the point directly opposite to starting point he should head the boat at an angle of `30^(@)` with AB or `90^(@)+30^(@)=120^(@)` with the river flow.
b. Time taken by the boatman to corss the river w=width of river =4km
`v_(br)=4`km/h and `theta=30^(@)`
`:.t=4/(4cos 30^(@))=2/(sqrt(3))h`
c.For shortest time `theta=0^(@)` and `t_("min")=w/(v_(br)cos 0^(@))=4/4=1h`
Hence he should head his boat perpendicular to the river current for crossing the river in shortest time and this shortest time is 1 h
d. `t=t_(CD)+t_(DC)`

or `t=(CD)/(v_(br)-v_(r))+(DC)/(v_(br)+v_(r))`
`=2/(4-2)+2/(4+2)`
`=1+1/3=4/3h`