The circular clock above shows a time of exactly 3:30. What is the val

Solution

Given:
• In the given figure, the circular clock shows a time of exactly 3:30

To find:
• The value of x

Approach and Working:

Method 1: Taking the reference time as 12:00 hrs
• If we consider the movement of the minute hand and hour hand separately, with respect to 12:00 hrs, we can calculate their individual displacement
• The minute hand of a clock covers a distance of 360° in 60 minutes

o Hence, it covers a distance of 6° in every minute

• The hour hand covers a distance of 360° in 12 hrs = 720 minutes

o Hence, it covers a distance of \(\frac{1}{2}\)° in every minute

Now, after the completion of every hour, minute hand comes to 12.
• Hence, at 3:30, the movement of the minute hand = 30 * 6° = 180°
From 12:00 hrs to 3:30, the hour hand moves for a time of 3.5 hrs = 210 minutes
• Hence, at 3:30, the movement of the hour hand = 210 * \(\frac{1}{2}\)° = 105°
• Therefore, the distance in between the minute and the hour hand = x = 180° - 105° = 75°

Method 2: Taking the reference time as 3:00 hrs
At 3:00 hrs, the minute hand will be at 12 and the hour hand will point towards 3, making a gap of 90° in between
• At 3:30 hrs, the minute hand will make a movement from 12 to exact opposite at 6, hence covering a distance of \(\frac{360°}{2}\) = 180°

o Hence, from 3, the displacement of minute hand = 180° - 90° = 90°

• At 3:30 hrs, the hour hand will move exactly in between 3 and 4, covering a distance of \(\frac{1}{2}\) * \(\frac{360°}{12}\) = 15°
• Therefore, the distance between the minute and the hour hand = x = 180° - 105° = 75°

Hence, the correct answer is option B.

Answer: B