Question

A boats man finds that he can save $6$ sec in crossing a river by quicker path, then by shortest path if the velocity of boat and river be respectively $17$ m/s and $8$ m/s, then river width is?

• A. $675$ m
• B. $765$ m
• C. $567$ m
• D. $657$ m

Answer 1

The correct option is B $765$ m

Let the width of the river be 'd' and the time taken to cross the river by the quickest and shortest path be $t_1$ and $t_2$ respectively.

Speed of boat = 17 $\frac{m}{s}$ = $v_1$ (say)

Speed of river = 8 $\frac{m}{s}$ = $v_2$ (say)

For crossing the river by the quickest path , the man has to move perpendicular to the direction of flow of the river (relative to water)

Therefore, $t_1$ = d/$v_1$

= $\frac{d}{17}$ seconds

For crossing the river by the shortest path , the motion of boat should be perpendicular to direction of river flow

Therefore, let the boat start moving at an angle $\theta$ to the perpendicular to the direction of river flow

$v_1$sin$\theta$ = 8 (for direction of motion to be perpendicular to direction of river flow)

$v_1$cos$\theta$ = 15

Therefore , $t_2$ = d/$v_2$

= $\frac{d}{15}$ seconds

According to question,

$t_2$ - $t_1$ = 6 seconds

$\frac{d}{15}$ - $\frac{d}{17}$ = 6 seconds

d = $\frac{6\ast 17\ast 15}{2}$ metres

Therefore , d = 765 metres