Question

The minute-hand and second-hand of a clock cross each other \underline{\hspace{1cm}} times between 09:15:00 AM and 09:45:00 AM on a day.

• A. 30
• B. 15
• C. 29
• D. 31

Hint:

Minute and second hand cross each other 59 times in one hour. In 30 minutes, they cross about half, i.e. ~29 times.

Answer 1

The Correct Option is C

Step 1: Hand Speeds.
\n- The second hand's speed is \(360^\circ/60 = 6^\circ/s\).
\n- The minute hand's speed is \(360^\circ/3600 = 0.1^\circ/s\). \n\n \n

Step 2: Relative Speed.
\nRelative speed of the hands is \(6 - 0.1 = 5.9^\circ/s\). \n\n \n

Step 3: Time Between Coincidences.
\nCoincidence occurs when the relative angular displacement is \(360^\circ\). \n\[\nT = \frac{360}{5.9} \approx 61.02 \, s\n\] \n\n \n

Step 4: Given Duration.
\nThe total time from 9:15 to 9:45 is 30 minutes, which equals \(1800 \, s\). \n\n \n

Step 5: Number of Coincidences.
\n\[\nN = \frac{1800}{61.02} \approx 29.49\n\] \nTherefore, the hands coincide **29 times** within this interval. \n\n \n\[\n\boxed{29}\n\]