A man wishes to cross a river of width 120 m by a motorboat. His rowing speed in still water is `3 m//s` and his maximum walking speed is `1m//s`. The river flows with velocity of `4 m//s`.
(a) Find the path which he should take to get to the point directly opposite to his starting point in the shortest time.
(b) Also, find the time which he takes to reach his destination.

Answer

A, B, C, D

Explanation

(a)

Time to cross the river
`t_1=120/(3cos theta)=40/cos theta=40sec theta`
Drift along the river `x=(4-3 sin theta)(40/cos theta)`
`=(16 sec theta-120 tan theta)`
To reach directly oppsite, this drift will be
covered by walking speed.
Time taken in this,
`t_2=(160 sec theta-120 tan theta)/1`
`=160 sec theta-120 tan theta`
`:.` Total time taken
`t=t_1+t_2=(200 sec theta-120 tan theta)`
For t to be minimum, `(dt)/(d theta)=0`
or `200 sec theta tan theta-120 sec^2 theta=0` or `theta=sin^-(3//5)` (b) `t_(min)=200 sec theta-120 tan theta ("where", sin theta=3/5)`
`=200xx5/4-120xx3/4`
`=250-90=160s=2 "min"40s`