From a point `A` on bank of a channel with still water a person must get to a point `B` on the opposite bank.All the distances are shown in figure.The person uses a boat to travel across the channel and then walks along the bank of point `B`.The velocity of the boat is `v_(1)` and the velocity of the walking person is `v_(2)`.Prove that the fastest way for the person to get from `A` to `B` is to select the angles `alpha_(1)` and `alpha_(2)` in such a manner that

Options

A. `(sin alpha_(1))/(sinalpha_(2))=v_(2)/v_(1)`
B. `(sin alpha_(1))/(sinalpha_(2))=v_(1)/v_(2)`
C. `(cos alpha_(1))/(cos alpha_(2))=v_(2)/v_(1)`
D. `(cos alpha_(2))/(cos alpha_(1))=v_(1)/v_(2)`

Answer

A

Explanation

`S_(1)=x/(sin alpha_(1)),S_(2)=(d-x)/(sin alpha_(2)),t_(1)=S_(1)/v_(1)=x/(v_(1)sin alpha_(1))`
and `t_(2)=S_(2)/v_(2)=(d-x)/(v_(2)sin alpha_(2))`
`t=x[1/(v_(1)sin alpha_(1))-1/(v_(2)sin alpha_(2))]+d/(v_(2)sin alpha_(2))`
For `t` be minimum `(dt)/(dx)=0` or
`1/(v_(1)sin alpha_(1))=1/(v_(2)sin alpha_(2)),v_(1)/v_(2)=sin alpha_(2)/sin alpha_(1)`