Two sea trawlers left a sea port simultaneously in two mutually perpendicular directions. Half an hour later, the shortest distance between them was 17 km, and another 15 minutes later, one sea trawler was 10.5 km farther from the origin than the other. Find the speed of each sea trawler. (SNAP, 2008)

### Concept & Pythagorean Theorem When two objects travel in mutually perpendicular directions from the same starting point, the shortest distance between them forms the hypotenuse of a right-angled triangle, where the distances traveled by each form the perpendicular sides. $$ \text{Distance} = \text{Speed} \times \text{Time} $$ $$ c^2 = a^2 + b^2 $$ ### Step-by-Step Solution 1. Let the speeds of the two trawlers be $x$ km/hr and $y$ km/hr respectively. 2. Distance traveled in half an hour (0.5 hr): $0.5x$ and $0.5y$. 3. The shortest distance between them is the hypotenuse: $$ \sqrt{(0.5x)^2 + (0.5y)^2} = 17 $$ $$ 0.25x^2 + 0.25y^2 = 289 $$ $$ x^2 + y^2 = 1156 $$ 4. Another 15 minutes later, total time = $30 + 15 = 45$ minutes = $0.75$ hours. 5. Distance traveled in 0.75 hours: $0.75x$ and $0.75y$. 6. One is 10.5 km farther than the other: $$ 0.75x - 0.75y = 10.5 \quad \text{(Assuming } x > y \text{)} $$ $$ x - y = \frac{10.5}{0.75} = 14 $$ 7. We have a system of two equations: $x^2 + y^2 = 1156$ and $x - y = 14$. 8. Rather than solving quadratics, substitute the options into these simplified equations. - Option (a): $30^2 + 16^2 = 900 + 256 = 1156$. And $30 - 16 = 14$. This fits perfectly. ### Exam Strategy & Shortcut Instead of solving the quadratic equation $(y + 14)^2 + y^2 = 1156$, use option elimination based on the derived condition $x - y = 14$. - (a) $30 - 16 = 14$ (Matches) - (b) $24 - 18 = 6$ (Fails) - (c) $22 - 20 = 2$ (Fails) - (d) $36 - 18 = 18$ (Fails) Only option (a) satisfies the difference in speeds. ### Common Pitfall A common mistake is incorrectly converting 45 minutes into hours (e.g., using 0.45 instead of 0.75), which completely ruins the second equation. ### Final Answer Therefore, the correct answer is **16 km/hr, 30 km/hr**.