Question 1

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…

Accepted Answer

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. Distance_{diameter} = 2r Distance_{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: (t/14) = (\pi r/2r) • t = 14 × (\pi/2) = 7\pi • Using \pi \approx (22/7): t = 7 × (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 (\pi r/2r) = (\pi/2) \approx (22/7/2) = (11/7). Multiply the usual time by this ratio to get the new time: 14 × (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.

Question 2

A skating champion moves along the circumference of a circle of radius 28 m in 44 sec. How many seconds will it take her to move along the perimeter of a hexagon of side 48 m?

Accepted Answer

Concept & Speed, Distance, and Time: This problem requires equating the speed of the skater on two different geometric paths. First, calculate the speed using the circular path, then use that speed to find the time taken for the hexagonal path. Speed = \frac{Distance}{Time} Step-by-Step Solution: • Step 1: Find the speed on the circular track. • Radius of the circle, r = 28 m. • Perimeter (circumference) of the circle = 2\pi r = 2 × (22/7) × 28. • Circumference = 2 × 22 × 4 = 176 m. • Time taken for the circle = 44 seconds. • Speed = \frac{176 m}{44 s} = 4 m/s. • Step 2: Find the time for the hexagonal track. • Side of the regular hexagon, a = 48 m. • Perimeter of the hexagon = 6 × a = 6 × 48 = 288 m. • Time taken = \frac{Distance}{Speed} = (288/4). • Time = 72 seconds. Exam Strategy & Shortcut: Instead of calculating the intermediate speed fully, you can set up a ratio since speed is constant: \frac{Time_1}{Distance_1} = \frac{Time_2}{Distance_2} (44/2 × \frac{22){7} × 28} = (T/6 × 48) \Rightarrow (44/176) = (T/288) \Rightarrow (1/4) = (T/288) \Rightarrow T = 72. Common Pitfall: A frequent mistake is using the formula for the area instead of the perimeter for the tracks. Always associate "moving along the circumference/perimeter" with boundary length, not internal area. Final Answer: Therefore, the correct answer is 72.

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