Kumbhakarna starts sleeping between 1 am and 2 am and he wakes up when his watch shows such a time that the two hands (i.e., hour-hand and minute-hand) interchange the respective places. He wakes up between 2 am and 3 am on the same night.How long does he sleep?

Options

A. 55 3/13 min
B. 110 10/13 min
C. 54 6/13 min
D. none of these

Answer

To solve the problem, we need to determine how long Kumbhakarna sleeps based on the positions of the hour and minute hands of the clock when he wakes up.

### Step-by-Step Solution:

1. **Understanding the Problem**:
Kumbhakarna starts sleeping between 1 AM and 2 AM and wakes up when the hour and minute hands of the clock interchange their positions. This means that if the hour hand is at a certain position, the minute hand will be at the position of the hour hand when he wakes up.

2. **Position of the Hands**:
The hour hand moves at a rate of 0.5 degrees per minute (360 degrees in 12 hours = 30 degrees per hour, so 30 degrees/60 minutes = 0.5 degrees per minute). The minute hand moves at a rate of 6 degrees per minute (360 degrees in 60 minutes = 6 degrees per minute).

3. **Let T be the time Kumbhakarna sleeps**:
If he sleeps for T minutes, then:
- The hour hand will move \(0.5T\) degrees.
- The minute hand will move \(6T\) degrees.

4. **Setting up the Equation**:
When Kumbhakarna wakes up, the positions of the hour and minute hands are interchanged. If he wakes up at a time between 2 AM and 3 AM, we can express the positions of the hands as follows:
- The hour hand's position when he wakes up is \(30 + 0.5T\) degrees (30 degrees for 1 hour + the additional movement during T minutes).
- The minute hand's position when he wakes up is \(6T\) degrees.

Since they interchange their positions:
\[ 30 + 0.5T = 6T \]

5. **Solving the Equation**:
Rearranging the equation:
\[ 30 = 6T - 0.5T \]
\[ 30 = 5.5T \]
\[ T = \frac{30}{5.5} = \frac{300}{55} = \frac{60}{11} \text{ minutes} \]

6. **Converting to Minutes and Seconds**:
To convert \( \frac{60}{11} \) minutes into minutes and seconds:
- \( 60 \div 11 = 5 \) minutes (whole part)
- Remainder: \( 60 - (5 \times 11) = 5 \) seconds
- So, \( T \approx 5 \) minutes and \( 27.27 \) seconds.

7. **Final Answer**:
Kumbhakarna sleeps for approximately \( 5 \) minutes and \( 27.27 \) seconds.