In order to reach his office on time, Mr. Roy goes through the middle passage of a round fort which he takes 14 minutes to pass through. However, on a certain day, due to repairs, the straight road being blocked, he had to take the roundabout way as a result of which he reached his office late. How late was he?

### Concept & Circular Motion The middle passage of a round fort represents its diameter. The roundabout way represents its semicircular arc. The time taken is directly proportional to the distance traveled if the speed remains constant. $$ \text{Distance}_{\text{diameter}} = 2r $$ $$ \text{Distance}_{\text{semicircle}} = \pi r $$ ### Step-by-Step Solution * Let the radius of the round fort be $r$. * Distance through the middle passage (diameter) = $2r$. * Time taken for distance $2r$ = 14 minutes. * Distance through the roundabout way (semicircle) = $\pi r$. * Let the time taken for the roundabout way be $t$. * Since speed is constant, the ratio of times equals the ratio of distances: $ \frac{t}{14} = \frac{\pi r}{2r} $ * $ t = 14 \times \frac{\pi}{2} = 7\pi $ * Using $\pi \approx \frac{22}{7}$: $t = 7 \times \frac{22}{7} = 22$ minutes. * The time he was late = Time taken for roundabout way - Usual time * Late by = $22 - 14 = 8$ minutes. ### Exam Strategy & Shortcut The ratio of the length of a semicircle to its diameter is $\frac{\pi r}{2r} = \frac{\pi}{2} \approx \frac{22/7}{2} = \frac{11}{7}$. Multiply the usual time by this ratio to get the new time: $14 \times \frac{11}{7} = 22$ mins. Difference = $22 - 14 = 8$ mins. ### Common Pitfall A common error is confusing the "roundabout way" with the full circumference. Since he crosses the fort to reach his office on the other side, he only travels half the circumference (a semicircle). ### Final Answer Therefore, the correct answer is **8 min**.