Multiple choice

A boatman finds that he can save $6 s$ in crossing a river by the quickest path than by the shortest path. If the velocity of the boat and the river be, respectively, $17 ms^{-1}$ and $8 ms^{-1}$, find the river width.

  1. $d=765m$
  2. $d=865m$
  3. $d=965m$
  4. $d=1065m$
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A Correct answer
Explanation

Let d be the width, v=17, u=8. Quickest path time t1 = d / sqrt(v^2 - u^2) = d / sqrt(289 - 64) = d / sqrt(225) = d / 15. Shortest path time t2 = d / v = d / 17. The difference t2 - t1 = 6. So d/15 - d/17 = 6. (17d - 15d) / 255 = 6, 2d = 1530, d = 765 m.

AI explanation

Using the concept of relative velocity in rivers, the quickest path involves pointing the boat perpendicular to the bank, taking a time of d/17 seconds, where d is the river width. The shortest path requires the boat's resultant velocity to be perpendicular to the bank, meaning its upstream component cancels the river's 8 m/s current, giving a perpendicular velocity of sqrt(17^2 - 8^2) = sqrt(225) = 15 m/s; this path takes d/15 seconds. The problem states the difference in time is 6 seconds, so (d/15) - (d/17) = 6, which simplifies to 2d/255 = 6. Solving for d gives d = 765 meters. The result is d=765m.

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